Sets, Relations, and Functions

Contents

Define and Classify the Set 1

List various Properties of Sets 2

Recall the following Set Operations: Union, Intersection, Disjoint etc 3

Recall Cardinality and Inclusion-Exclusion Theorem 5

Define and classify the Relation 7

Find Domain, Co-domain, and Range of a given Relation 9

Find Relation Matrix and draw its Directed Graph 10

Describe the Partial Order Relation 11

Verify that Relation R is Partial Order 12

Describe the Closure of a Relation 13

Define and classify the Functions 14

Find the value of Domain and Composite Function 15

Differentiate between Relation and Function 17

Recall the concept of Mathematical Induction 19

Apply the concept of Mathematical Induction to verify the given Statement 20

Define and Classify the Set

In discrete mathematics, a set is a collection of distinct elements. It is a fundamental concept used to represent and study collections of objects. The elements in a set can be anything, such as numbers, letters, or even other sets.

A set is defined by listing its elements inside curly braces {}. For example, {1, 2, 3} is a set containing the elements 1, 2, and 3. If an element appears multiple times in a set, it is considered only once as duplicates are not allowed in a set.

Sets can be classified into various types based on their characteristics and properties. Here are some common classifications of sets:

1. Finite Set: A finite set is a set that contains a specific number of elements. For example, {1, 2, 3} is a finite set with three elements.
2. Infinite Set: An infinite set is a set that contains an unlimited number of elements. For example, the set of natural numbers {1, 2, 3, …} is an infinite set.
3. Empty Set: Also known as the null set, the empty set is a set that does not contain any elements. It is denoted by the symbol Ø or {}.
4. Singleton Set: A singleton set is a set that contains exactly one element. For example, {5} is a singleton set.
5. Subset: A subset is a set that contains only elements that are also contained in another set. For example, {1, 2} is a subset of {1, 2, 3}.
6. Power Set: The power set of a set is the set of all possible subsets of that set. For example, the power set of {1, 2} is {{}, {1}, {2}, {1, 2}}.
7. Universal Set: The universal set is a set that contains all the elements under consideration in a given context. It is often denoted by the symbol U.

Sets in discrete mathematics have various applications in different areas, including combinatorics, logic, graph theory, and computer science. They provide a foundation for understanding and reasoning about collections of objects and their relationships.

List various Properties of Sets

Sets have several properties that govern their behavior and relationships. Here are some important properties of sets:

1. Cardinality: The cardinality of a set is the number of elements it contains. It can be finite or infinite.
2. Equality: Two sets are considered equal if they have exactly the same elements. The order of elements or their repetitions do not matter in determining set equality.
3. Subset: A set A is said to be a subset of another set B if every element of A is also an element of B. If A is a subset of B, it is denoted as A ⊆ B.
4. Proper Subset: A set A is said to be a proper subset of another set B if A is a subset of B but A is not equal to B. If A is a proper subset of B, it is denoted as A ⊂ B.
5. Universal Set: The universal set is the set that contains all the elements under consideration in a given context. It serves as a reference for defining subsets and complements.
6. Empty Set: Also known as the null set, the empty set is a set that does not contain any elements. It is denoted by the symbol Ø or {}.
7. Union: The union of two sets A and B is the set that contains all the elements that are in A, in B, or in both. It is denoted as A ∪ B.
8. Intersection: The intersection of two sets A and B is the set that contains all the elements that are common to both A and B. It is denoted as A ∩ B.
9. Complement: The complement of a set A with respect to a universal set U is the set that contains all the elements in U that are not in A. It is denoted as A’.
10. Disjoint Sets: Two sets are said to be disjoint if they have no common elements, i.e., their intersection is the empty set.
11. Power Set: The power set of a set is the set of all possible subsets of that set. It includes the empty set and the set itself.
12. Cartesian Product: The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. It is denoted as A × B.

These properties and operations provide the foundation for set theory and its applications in various areas of mathematics and computer science.

Recall the following Set Operations: Union, Intersection, Disjoint etc

Here are the definitions of the set operations you mentioned:

1. Union: The union of two sets A and B, denoted as A ∪ B, is the set that contains all the elements that are in either set A or set B, or in both. In other words, it is the combination of all unique elements from both sets.
2. Intersection: The intersection of two sets A and B, denoted as A ∩ B, is the set that contains all the elements that are common to both sets. In other words, it is the set of elements that belong to both A and B.
3. Disjoint Sets: Two sets A and B are said to be disjoint if they have no common elements, i.e., their intersection is the empty set (∅). In other words, A and B do not share any elements.
4. Complement: The complement of a set A, denoted as A’, is the set of all elements that are in the universal set but not in A. It represents the elements that are not part of A.
5. Difference: The difference of two sets A and B is the set of all elements that are in A but not in B. It is denoted by A \ B.

These set operations are fundamental to set theory and provide a way to manipulate and analyze sets. They are used to define relationships between sets, combine sets, find common elements, and differentiate elements. These operations can be applied to any type of set, including finite and infinite sets.

Here are some examples of set operations using specific sets:

Let’s consider two sets:

Set A: {1, 2, 3, 4}

Set B: {3, 4, 5, 6}

1. Union (A ∪ B):
   • A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6}
   • A ∪ B = {1, 2, 3, 4, 5, 6}
   • The union of sets A and B contains all the unique elements from both sets: {1, 2, 3, 4, 5, 6}.
2. Intersection (A ∩ B):
   • A ∩ B = {1, 2, 3, 4} ∩ {3, 4, 5, 6}
   • A ∩ B = {3, 4}
   • The intersection of sets A and B contains the common elements between the two sets: {3, 4}.
3. Disjoint Sets:
   • Sets A and B are not disjoint because they have common elements (3 and 4).
4. Complement (A’):
   • Assuming there is a universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
   • A’ represents the elements in the universal set that are not in A.
   • A’ = {5, 6, 7, 8, 9, 10}
   • The complement of set A contains the elements from the universal set that are not in A.

5. Here’s an example that illustrates the difference between two sets:

Set A: {1, 2, 3, 4}

Set B: {3, 4, 5, 6}

1. Difference (A – B):
   • A – B = {1, 2, 3, 4} – {3, 4, 5, 6}
   • A – B = {1, 2}
   • The difference between sets A and B (A – B) contains the elements that are in set A but not in set B.
2. Difference (B – A):
   • B – A = {3, 4, 5, 6} – {1, 2, 3, 4}
   • B – A = {5, 6}
   • The difference between sets B and A (B – A) contains the elements that are in set B but not in set A.

In this example, when we calculate A – B, we find that the elements {1, 2} are in set A but not in set B. On the other hand, when we calculate B – A, we find that the elements {5, 6} are in set B but not in set A.

The difference operation allows us to compare two sets and identify the elements that are unique to each set. It provides a way to find elements that are present in one set but not in the other.

Recall Cardinality and Inclusion-Exclusion Theorem

Here’s a recall of Cardinality and the Inclusion-Exclusion Theorem:

1. Cardinality: In set theory, the cardinality of a set is the number of elements it contains. It measures the “size” or “count” of a set. The cardinality of a finite set can be determined by counting its elements, while the cardinality of an infinite set can be determined using various mathematical techniques. The cardinality of a set A is denoted as |A|.
2. Inclusion-Exclusion Theorem: The Inclusion-Exclusion Theorem is a principle used to calculate the cardinality of the union of multiple sets. It provides a formula for finding the size of the union of sets, taking into account the overlap between the sets.
   The theorem states that for any finite sets A1, A2, …, An, the cardinality of their union can be calculated as follows:
   |A1 ∪ A2 ∪ … ∪ An| = |A1| + |A2| + … + |An| – |A1 ∩ A2| – |A1 ∩ A3| – … – |An-1 ∩ An| + |A1 ∩ A2 ∩ A3| + … + (-1)⁽ⁿ⁺¹⁾ |A1 ∩ A2 ∩ … ∩ An|

In other words, the cardinality of the union is obtained by summing the cardinalities of the individual sets, subtracting the cardinalities of the pairwise intersections, adding back the cardinalities of the triple intersections, and so on. The alternating signs (+/-) in the formula account for the overcounting of elements in the intersections.

The Inclusion-Exclusion Theorem is useful for solving counting problems and determining the size of complicated unions of sets. It provides a systematic approach to handle overlapping sets and avoid double counting or omitting elements when calculating cardinalities.

Here are a couple of examples that demonstrate the application of the Inclusion-Exclusion Theorem:

Example 1:

Consider three sets:

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

C = {4, 5, 6, 7}

We want to find the cardinality of the union of these sets: |A ∪ B ∪ C|.

Using the Inclusion-Exclusion Theorem, we can calculate it as follows:

|A ∪ B ∪ C| = |A| + |B| + |C| – |A ∩ B| – |A ∩ C| – |B ∩ C| + |A ∩ B ∩ C|

Substituting the values:

|A ∪ B ∪ C| = 4 + 4 + 4 – 2 – 1 – 2 + 1

|A ∪ B ∪ C| = 8

Therefore, the cardinality of the union of sets A, B, and C is 8.

Example 2:

Consider two sets:

A = {1, 2, 3, 4, 5}

B = {3, 4, 5, 6, 7}

We want to find the cardinality of the union of these sets: |A ∪ B|.

Using the Inclusion-Exclusion Theorem, we can calculate it as follows:

|A ∪ B| = |A| + |B| – |A ∩ B|

Substituting the values:

|A ∪ B| = 5 + 5 – 3

|A ∪ B| = 7

Therefore, the cardinality of the union of sets A and B is 7.

These examples demonstrate how the Inclusion-Exclusion Theorem can be used to calculate the size of the union of sets, taking into account the overlapping elements and avoiding double counting.

Define and classify the Relation

In mathematics, a relation is a connection or association between two or more elements. It defines a relationship or correspondence between elements from different sets. Relations are used to study the interactions, dependencies, and comparisons between objects or entities.

Relations can be classified based on various properties and characteristics. Here are some common classifications of relations:

1. Binary and Multivariate Relations: Binary relations involve two sets of elements, denoted as A and B, and establish a connection between elements from A and B. For example, a binary relation can be represented as R ⊆ A × B, where R is the relation between A and B. Multivariate relations involve more than two sets of elements and establish connections among them.
2. Reflexive, Symmetric, and Transitive Relations: These classifications are based on the properties exhibited by the relation.
   • Reflexive Relation: A relation R on a set A is reflexive if every element in A is related to itself. In other words, for every element a in A, (a, a) belongs to R.
   • Symmetric Relation: A relation R on a set A is symmetric if for every pair of elements (a, b) in R, the pair (b, a) is also in R. It implies that the relation is bidirectional or symmetric with respect to the elements involved.
   • Transitive Relation: A relation R on a set A is transitive if for every triplet (a, b, c) in R, if (a, b) and (b, c) are in R, then (a, c) is also in R. It means that if there is a chain of relations between elements, the relation extends to the final element as well.
3. Equivalence Relations: An equivalence relation is a special type of relation that satisfies three properties: reflexivity, symmetry, and transitivity. Equivalence relations are used to define equivalence classes and partition a set into subsets with similar properties.
4. Partial Order and Total Order Relations: Partial order relations, also known as partial orders or partial orders, are relations that are reflexive, antisymmetric, and transitive. They define a partial ordering or a partial ranking among elements. Total order relations, also known as total orders or total orders, are partial order relations where every pair of elements is comparable. In other words, for any two elements a and b in the set, either (a, b) or (b, a) belongs to the relation.

These classifications provide a way to analyze and study the properties and behavior of relations in mathematics. Each classification serves a specific purpose and helps in understanding the nature and characteristics of the relation under consideration.

Here are some examples of relations with their classifications:

1. Binary Relation:

Let A = {1, 2, 3} and B = {4, 5, 6}. The relation R = {(1, 4), (2, 5), (3, 6)} is a binary relation between sets A and B.

1. Reflexive Relation:

Let A = {1, 2, 3}. The relation R = {(1, 1), (2, 2), (3, 3)} is a reflexive relation on set A since each element is related to itself.

1. Symmetric Relation:

Let A = {1, 2, 3}. The relation R = {(1, 2), (2, 1), (2, 3), (3, 2)} is a symmetric relation on set A since for every pair (a, b) in R, the pair (b, a) is also in R.

1. Transitive Relation:

Let A = {1, 2, 3}. The relation R = {(1, 2), (2, 3), (1, 3)} is a transitive relation on set A since for every triplet (a, b, c) in R, if (a, b) and (b, c) are in R, then (a, c) is also in R.

1. Equivalence Relation:

Let A = {1, 2, 3, 4, 5}. The relation R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 3), (3, 1)} is an equivalence relation on set A since it is reflexive, symmetric, and transitive.

1. Partial Order Relation:

Let A = {1, 2, 3, 4}. The relation R = {(1, 1), (1, 2), (2, 2), (2, 4), (3, 3), (4, 4)} is a partial order relation on set A since it is reflexive, antisymmetric, and transitive.

These examples illustrate different types of relations and their classifications based on the properties they satisfy.

Find Domain, Co-domain, and Range of a given Relation

To find the domain, co-domain, and range of a given relation, you need to understand the elements involved in the relation and how they are related.

Here’s how you can determine the domain, co-domain, and range:

1. Domain: The domain of a relation refers to the set of all input or starting elements of the relation. In other words, it represents the values from which the relation originates.

To find the domain, identify all the elements that appear as the first element of ordered pairs in the relation. Collect these elements to form the domain set.

1. Co-domain: The co-domain of a relation refers to the set of all possible output or target elements of the relation. It represents the set of values that the elements in the domain can be related to.

To determine the co-domain, examine all the possible target elements of the relation. Collect these elements to form the co-domain set.

1. Range: The range of a relation refers to the set of all actual output or target elements of the relation. It represents the values that are actually related to the elements in the domain.

To find the range, examine the second elements of the ordered pairs in the relation. Collect these elements to form the range set.

Here’s an example to illustrate the process:

Given relation: R = {(1, 2), (2, 3), (3, 4), (4, 5)}

Domain: The first elements in the ordered pairs are 1, 2, 3, and 4. Therefore, the domain is {1, 2, 3, 4}.

Co-domain: The possible target elements in the relation are 2, 3, 4, and 5. Therefore, the co-domain is {2, 3, 4, 5}.

Range: The second elements in the ordered pairs are 2, 3, 4, and 5. Therefore, the range is {2, 3, 4, 5}.

In this example, the domain, co-domain, and range of the given relation R are all the same, which is {1, 2, 3, 4}.

Find Relation Matrix and draw its Directed Graph

To find the relation matrix and draw the directed graph of a given relation, you need to analyze the ordered pairs that make up the relation.

Here’s how you can do it:

1. Relation Matrix: The relation matrix is a square matrix that represents the relation using 0s and 1s. The rows and columns of the matrix correspond to the elements in the domain and co-domain, respectively. If an element from the domain is related to an element from the co-domain, the corresponding entry in the matrix is 1; otherwise, it is 0.

To create the relation matrix, follow these steps:

• List down the distinct elements from the domain and co-domain.
• Place a 1 in the matrix entry corresponding to each ordered pair in the relation.
• Place a 0 in the matrix entry for all other pairs not present in the relation.

1. Directed Graph: The directed graph represents the relation visually. Each element from the domain and co-domain is represented as a vertex, and the directed edges indicate the relation between them. An edge is drawn from an element in the domain to an element in the co-domain if the ordered pair is present in the relation.

To draw the directed graph, follow these steps:

• Represent each element from the domain and co-domain as a vertex.
• Draw a directed edge from the domain vertex to the co-domain vertex for each ordered pair in the relation.

Let’s work through an example:

Given relation: R = {(1, 2), (2, 3), (3, 4), (4, 1)}

Domain: {1, 2, 3, 4}

Co-domain: {1, 2, 3, 4}

1. Relation Matrix:

The relation matrix for the given relation is as follows:
| 1 | 2 | 3 | 4 |

1 | 0 | 1 | 0 | 0 |

2 | 0 | 0 | 1 | 0 |

3 | 0 | 0 | 0 | 1 |

4 | 1 | 0 | 0 | 0 |

1. Directed Graph:

The directed graph representing the given relation is as follows:
1 –> 2

↑ ↓

4 <– 3

In this example, the relation matrix and directed graph demonstrate the relation between the elements 1, 2, 3, and 4 according to the given relation R.

Describe the Partial Order Relation

A partial order relation, also known as a partial order or partial ordering, is a binary relation that satisfies three key properties: reflexivity, antisymmetry, and transitivity. It is a way of establishing a partial ranking or ordering among elements in a set.

1. Reflexivity: In a partial order relation, every element is related to itself. This means that for every element a in the set, (a, a) belongs to the relation. In other words, the relation is reflexive.
2. Antisymmetry: A partial order relation is antisymmetric, which means that if two elements are related in one direction, they cannot be related in the opposite direction. More formally, for any elements a and b in the set, if (a, b) belongs to the relation and (b, a) also belongs to the relation, then a must be equal to b. This property ensures that there are no strict cycles or loops in the ordering.
3. Transitivity: The transitive property states that if two elements are related and the second element is related to a third element, then the first element is also related to the third element. In terms of partial order relations, if (a, b) and (b, c) belong to the relation, then (a, c) must also belong to the relation. This property ensures that the ordering is consistent and transitive.

Together, these three properties define a partial order relation. It provides a way to compare and order elements in a set based on certain criteria or attributes. However, it does not necessarily establish a total ordering, meaning that not all elements need to be comparable or related to each other.

A common example of a partial order relation is the “less than or equal to” relation (≤) on the set of real numbers. In this relation, any number is related to itself, if one number is less than or equal to another, they cannot be strictly greater than each other, and if one number is less than or equal to another and the second number is less than or equal to a third number, then the first number is also less than or equal to the third number.

Verify that Relation R is Partial Order

To verify whether a given relation R is a partial order, we need to check if it satisfies the three properties of reflexivity, antisymmetry, and transitivity. Let’s consider the relation R and examine each property:

1. Reflexivity: For a relation to be reflexive, every element in the set must be related to itself. In other words, (a, a) must belong to the relation for all elements a in the set.
2. Antisymmetry: Antisymmetry requires that if (a, b) and (b, a) belong to the relation, then a must be equal to b. In other words, there should be no distinct elements a and b for which both (a, b) and (b, a) are in the relation.
3. Transitivity: Transitivity states that if (a, b) and (b, c) belong to the relation, then (a, c) must also belong to the relation. This ensures that the relation is consistent and forms a chain of relationships.

By checking these three properties, we can determine if the given relation R is a partial order.

Let’s consider another example for the relation R:

R = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 3)}

To verify if R is a partial order relation, let’s examine the properties:

1. Reflexivity: For every element x in the set, (x, x) must belong to the relation R. In this case, (1, 1), (2, 2), and (3, 3) are present in R, satisfying the reflexivity property.
2. Antisymmetry: If (a, b) and (b, a) are in the relation R, then a must be equal to b. In this case, we don’t have any distinct elements a and b for which both (a, b) and (b, a) exist. So, the antisymmetry property is satisfied.
3. Transitivity: If (a, b) and (b, c) belong to R, then (a, c) should also be in R. Looking at the pairs in R, we can see that (1, 2) and (2, 3) are present, and as a result, (1, 3) is also in R. Thus, the transitivity property is satisfied.

Since the relation R satisfies the properties of reflexivity, antisymmetry, and transitivity, we can conclude that R is a partial order relation.

Describe the Closure of a Relation

The closure of a relation is a concept in mathematics that extends or completes a given relation to include additional elements based on certain properties or rules. It aims to create a new relation that encompasses all the desired elements and preserves specific properties of the original relation.

There are different types of closures, such as reflexive closure, symmetric closure, and transitive closure, depending on the properties being preserved. Here, we will focus on the transitive closure, as it is commonly discussed in relation to the closure of a relation.

The transitive closure of a relation R over a set A is the smallest transitive relation that contains R. In other words, it is the additional set of ordered pairs that need to be added to R to make it transitive.

To find the transitive closure of a relation, you can follow these steps:

1. Start with the given relation R.
2. Add any missing pairs that are required to make R transitive. For each pair (a, b) in R, check if there exists a pair (b, c) in R. If so, add the pair (a, c) to the transitive closure.
3. Repeat step 2 until no more pairs can be added.
4. The resulting set of ordered pairs is the transitive closure of the relation R.

For example, consider the relation R = {(1, 2), (2, 3)}:

To find the transitive closure:

• (1, 2) is already in R, and there is no pair (2, c) in R, so we add (1, 3) to the transitive closure.
• The transitive closure now becomes R* = {(1, 2), (2, 3), (1, 3)}.

In this case, (1, 3) needed to be added to make R transitive, and the resulting transitive closure is R* = {(1, 2), (2, 3), (1, 3)}.

The transitive closure ensures that the relation includes all the necessary pairs to establish transitivity, allowing for a more complete understanding of the relationships between the elements in the set.

Define and classify the Functions

In discrete mathematics, a function is a mathematical relation between two sets that assigns each element from the first set (called the domain) to a unique element in the second set (called the codomain). It specifies a rule or mapping that associates inputs with corresponding outputs.

A function can be defined as follows:

Let A and B be two sets. A function f from A to B, denoted as f: A -> B, is a relation that assigns each element ‘a’ in set A to a unique element ‘b’ in set B. For each ‘a’ in A, there exists exactly one ‘b’ in B such that (a, b) belongs to the function f.

Classification of Functions:

1. One-to-One (Injective) Function: A function is said to be one-to-one or injective if each element in the domain maps to a unique element in the codomain. In other words, no two distinct elements in the domain map to the same element in the codomain.
2. Onto (Surjective) Function: A function is said to be onto or surjective if each element in the codomain has at least one pre-image in the domain. In other words, every element in the codomain is mapped to by at least one element in the domain.
3. Bijective Function: A function is said to be bijective if it is both one-to-one and onto. It means that each element in the domain maps to a unique element in the codomain, and every element in the codomain has exactly one pre-image in the domain.
4. Many-to-One Function: A function is said to be many-to-one if multiple elements in the domain map to the same element in the codomain. In this case, the function is not injective.
5. One-to-Many Function: A function is said to be one-to-many if an element in the domain maps to multiple elements in the codomain. In this case, the function is not well-defined.
6. Identity Function: An identity function is a function where each element in the domain maps to itself. It is both one-to-one and onto.

These classifications help us understand the properties and behavior of functions in discrete mathematics and their role in various mathematical applications.

Find the value of Domain and Composite Function

To find the value of the domain and composite function, we need to understand the concepts and definitions involved.

1. Domain of a Function:

The domain of a function is the set of all possible input values or arguments for which the function is defined. It represents the valid inputs that can be used to evaluate the function. The domain can vary depending on the specific function being considered.

For example, let’s consider the function f(x) = 2x. In this case, the domain of the function is all real numbers because we can input any real number value for ‘x’ and obtain a valid output.

1. Composite Function:

A composite function is a function that is formed by combining two or more functions. The output of one function is used as the input for another function. The composite function is obtained by performing the operations of the individual functions in a specific order.

For example, let’s consider two functions: f(x) = 2x and g(x) = x + 3. The composite function (f ∘ g)(x) represents the composition of these two functions, where the output of g(x) is used as the input for f(x).

The composite function can be calculated as follows:

(f ∘ g)(x) = f(g(x))

= f(x + 3)

= 2(x + 3)

= 2x + 6

Now, let’s find the value of the composite function for a specific input, say x = 4:

(f ∘ g)(4) = 2(4) + 6

= 8 + 6

= 14

So, the value of the composite function (f ∘ g)(4) is 14.

In summary, the domain of a function refers to the set of valid input values, and the composite function is formed by combining two or more functions. The value of the composite function can be calculated by substituting the input value into the composed expression and simplifying it.

Differentiate between Relation and Function

Here is a tabular comparison between a relation and a function:

Property	Relation	Function
Definition	A relation is a set of ordered pairs.	A function is a special type of relation
	Each ordered pair represents a connection	where each input has a unique output.
	between elements of two sets.
Input and	A relation can have multiple outputs	A function has exactly one output for each
Output	for a given input.	input.
Representation	A relation can be represented as a set of	A function is often represented using the
	ordered pairs, a table, or a graph.	functional notation: f(x) or y = f(x).
Domain and	The domain and codomain of a relation	The domain and codomain of a function are
Codomain	can have overlapping or different sets.	well-defined and often the same.
Example	{(1, 2), (3, 4), (2, 1)}	f(x) = x2

In summary, a relation is a general concept that represents a connection between elements of two sets, whereas a function is a specific type of relation where each input has a unique output. A relation can have multiple outputs for a given input, while a function has exactly one output for each input. Relations can be represented in various forms, such as sets, tables, or graphs, whereas functions are often represented using functional notation.

Additionally, the domain and codomain of a relation can have overlapping or different sets, while the domain and codomain of a function are well-defined and often the same.

Recall the concept of Mathematical Induction

Mathematical induction is a proof technique used to establish that a given statement or property holds for all natural numbers (or a subset of natural numbers) by proving it for a specific base case and showing that if it holds for any particular value, it also holds for the next value. The concept of mathematical induction is based on the principle of well-ordering of natural numbers.

The process of mathematical induction typically involves three steps:

1. Base Step: The first step is to prove that the statement or property holds true for a specific starting value, often the smallest value in the set of natural numbers. This is called the base case.
2. Inductive Step: The second step is to assume that the statement is true for some arbitrary value, often denoted as ‘k’. This is called the inductive hypothesis. Using this assumption, the goal is to prove that the statement is also true for the next value, which is ‘k + 1’. This is called the inductive step.
3. Conclusion: The final step is to conclude that the statement holds true for all natural numbers by applying the principle of mathematical induction. This involves asserting that if the statement holds for the base case and if it holds for an arbitrary value ‘k’, then it must hold for the value ‘k + 1’. Therefore, by this reasoning, the statement is true for all natural numbers.

Mathematical induction is commonly used to prove statements about mathematical properties, sequences, series, and combinatorial problems. It provides a rigorous and systematic approach to establish the validity of a statement across all natural numbers, avoiding the need to verify it individually for each number.

Apply the concept of Mathematical Induction to verify the given Statement

Here are five examples where we can apply the concept of mathematical induction to verify the given statements:

Example 1: Prove that the sum of the first ‘n’ natural numbers is equal to n(n+1)/2.

Statement: 1 + 2 + 3 + … + n = n(n+1)/2

Step 1 (Base Step): For n = 1, the left-hand side (LHS) is 1, and the right-hand side (RHS) is (1)(1+1)/2 = 1. So, the statement holds true for the base case.

Step 2 (Inductive Step): Assume that the statement holds true for some arbitrary value ‘k’, i.e., 1 + 2 + 3 + … + k = k(k+1)/2.

Step 3 (Conclusion): Now, we need to prove that the statement also holds true for the next value ‘k+1’. Adding (k+1) to both sides of the assumed equation, we get:

1 + 2 + 3 + … + k + (k+1) = k(k+1)/2 + (k+1)

= (k² + k + 2k + 2) / 2

= (k² + 3k + 2) / 2

= (k+1)(k+2) / 2

This shows that the statement is also true for ‘k+1’. Therefore, by the principle of mathematical induction, the statement is true for all natural numbers.

Example 2: Prove that 2^n > n² for all positive integers n ≥ 5.

Statement: 2^n > n², for n ≥ 5.

Step 1 (Base Step): For n = 5, the LHS is 2^5 = 32, and the RHS is 5² = 25. So, the statement holds true for the base case.

Step 2 (Inductive Step): Assume that the statement holds true for some arbitrary value ‘k’, i.e., 2^k > k².

Step 3 (Conclusion): Now, we need to prove that the statement also holds true for the next value ‘k+1’. Multiplying both sides of the assumed equation by 2, we get:

2^(k+1) > 2k^2

We can observe that 2k^2 < (k+1)^2 for all k ≥ 5. Hence, we can write:

2^(k+1) > (k+1)^2

This shows that the statement is also true for ‘k+1’. Therefore, by the principle of mathematical induction, the statement is true for all n ≥ 5.

Similarly, you can apply mathematical induction to verify other statements by following the three steps mentioned above. Just ensure that the base case is true and that you properly apply the inductive hypothesis and the induction step to prove the statement for the next value.

Example 3: Prove that 3^n > n^3 for all positive integers n ≥ 2.

Statement: 3^n > n^3, for n ≥ 2.

Step 1 (Base Step): For n = 2, the LHS is 3^2 = 9, and the RHS is 2^3 = 8. So, the statement holds true for the base case.

Step 2 (Inductive Step): Assume that the statement holds true for some arbitrary value ‘k’, i.e., 3^k > k^3.

Step 3 (Conclusion): Now, we need to prove that the statement also holds true for the next value ‘k+1’. Multiplying both sides of the assumed equation by 3, we get:

3^(k+1) > 3k^3

We can observe that 3k^3 < (k+1)^3 for all k ≥ 2. Hence, we can write:

3^(k+1) > (k+1)^3

This shows that the statement is also true for ‘k+1’. Therefore, by the principle of mathematical induction, the statement is true for all n ≥ 2.

Example 4: Prove that 1 + 2 + 2^2 + … + 2^n = 2^(n+1) – 1 for all non-negative integers n.

Statement: 1 + 2 + 2^2 + … + 2^n = 2^(n+1) – 1

Step 1 (Base Step): For n = 0, the LHS is 1, and the RHS is 2^1 – 1 = 1. So, the statement holds true for the base case.

Step 2 (Inductive Step): Assume that the statement holds true for some arbitrary value ‘k’, i.e., 1 + 2 + 2^2 + … + 2^k = 2^(k+1) – 1.

Step 3 (Conclusion): Now, we need to prove that the statement also holds true for the next value ‘k+1’. Adding 2^(k+1) to both sides of the assumed equation, we get:

1 + 2 + 2^2 + … + 2^k + 2^(k+1) = 2^(k+1) – 1 + 2^(k+1)

= 2^(k+2) – 1

This shows that the statement is also true for ‘k+1’. Therefore, by the principle of mathematical induction, the statement is true for all non-negative integers n.

Example 5: Prove that the sum of the cubes of the first n natural numbers is equal to (n(n+1)/2)^2.

Statement: 1^3 + 2^3 + 3^3 + … + n^3 = (n(n+1)/2)^2

Step 1 (Base Step): For n = 1, the LHS is 1^3 = 1, and the RHS is (1(1+1)/2)^2 = 1^2 = 1. So, the statement holds true for the base case.

Step 2 (Inductive Step): Assume that the statement holds true for some arbitrary value ‘k’, i.e., 1^3 + 2^3 + 3^3 + … + k^3 = (k(k+1)/2)^2.

Step 3 (Conclusion): Now, we need to prove that the statement also holds true for the next value ‘k+1’. Adding (k+1)^3 to both sides of the assumed equation, we get:

1^3 + 2^3 + 3^3 + … + k^3 + (k+1)^3 = (k(k+1)/2)^2 + (k+1)^3

= [(k^2 + k)/2]^2 + (k+1)^3

= [(k^2 + k)^2 / 4] + (k+1)^3

By simplifying the RHS, we can show that it is equal to [(k^2 + 2k + 1)(k^2 + k + 2) / 4], which can be further simplified to [(k+1)(k+2)/2]^2. Hence, the statement is also true for ‘k+1’.

Therefore, by the principle of mathematical induction, the statement is true for all natural numbers.

In each of these examples, we have followed the three steps of mathematical induction: base step, inductive step, and conclusion, to verify the given statements for a range of values.