Equivalence relation on set is a relation which is reflexive, symmetric and transitive. A relation R, defined in a set A, is said to be an equivalence relation if and only if (i) R is reflexive, that is, aRa for all a ∈ A.(ii) R is symmetric, that is, aRb ⇒ bRa for all a, b ∈ A. (iii) R is transitive, that is aRb and bRc ⇒ aRc for all a, b, c ∈ A. The relation defined by … [Read more...]

## Transitive Relation on Set

What is transitive relation on set?Let A be a set in which the relation R defined.R is said to be transitive, if(a, b) ∈ R and (b, a) ∈ R ⇒ (a, c) ∈ R,That is aRb and bRc ⇒ aRc where a, b, c ∈ A.The relation is said to be non-transitive, if(a, b) ∈ R and (b, c) ∈ R do not imply (a, c ) ∈ R.For example, in the set A of natural numbers if the relation R be defined by ‘x less than … [Read more...]

## Anti-symmetric Relation on Set

What is anti-symmetric relation onset? Let A be a set in which the relation R defined. R is said to be anti-symmetric, if there exist elements, if aRb and bRa ⇒ a = b that is, (a, b) ∈ R and ((b, a) ∈ R ⇒ a = b. A relation R in A is not anti-symmetric, if there exist elements a, b ∈ A, a ≠ b such that aRb and bRa.For example, the relation defined by ‘x is less than or … [Read more...]

## Symmetric Relation on Set

Here we will discuss about the symmetric relation on set.Let A be a set in which the relation R defined. Then R is said to be a symmetric relation, if (a, b) ∈ R ⇒ (b, a) ∈ R, that is, aRb ⇒ bRa for all (a, b) ∈ R.Consider, for example, the set A of natural numbers. If a relation A be defined by “x + y = 5”, then this relation is symmetric in A, for a + b = 5 ⇒ b + a = 5 But … [Read more...]

## Reflexive Relation on Set

Reflexive relation on set is a binary element in which every element is related to itself.Let A be a set and R be the relation defined in it.R is set to be reflexive, if (a, a) ∈ R for all a ∈ A that is, every element of A is R-related to itself, in other words aRa for every a ∈ A.A relation R in a set A is not reflexive if there be at least one element a ∈ A such that (a, a) ∉ … [Read more...]

## Properties of Elements in Sets

The following properties of elements in sets are discussed here.If U be the universal set and A, B and C are any three finite sets then;1. If A and B are any two finite sets then n(A - B) = n(A) – n(A ∩ B) i.e. n(A – B) + n(A ∩ B) = n(A)2. If A and B are any two finite sets then n(A ∪ B) = n(A) + n(B) – n(A ∩ B)3. If A and B are any two finite sets then n(A ∪ B) = n(A) + n(B) … [Read more...]

## Proof of De Morgan’s Law

Here we will learn how to proof of De Morgan’s law of union and intersection.Definition of De Morgan’s law: The complement of the union of two sets is equal to the intersection of their complements and the complement of the intersection of two sets is equal to the union of their complements. These are called De Morgan’s laws.For any two finite sets A and B;(i) (A U B)' = A' ∩ … [Read more...]

## Laws of Algebra of Sets

Here we will learn about some of the laws of algebra of sets.1. Commutative Laws:For any two finite sets A and B;(i) A U B = B U A(ii) A ∩ B = B ∩ A2. Associative Laws:For any three finite sets A, B and C;(i) (A U B) U C = A U (B U C)(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)Thus, union and intersection are associative. 3. Idempotent Laws: For any finite set A; (i) A U A = A(ii) A ∩ … [Read more...]

## Examples on Venn Diagram

Solved examples on Venn diagram are discussed here. From the adjoining Venn diagram, find the following sets. (i) A(ii) B (iii) ξ (iv) A'(v) B'(vi) C'(vii) C - A (viii) B - C (ix) A - B (x) A ∪ B (xi) B ∪ C (xii) A ∩ C (xiii) B ∩ C(xiv) (B ∪ C)'(xv) (A ∩ B)'(xvi) (A ∪ B) ∩ C(xvii) A ∩ (B ∩ C) Answers for examples on Venn diagram are given below: (i) A= {1, 3, 4, 5} (ii) … [Read more...]

## Complement of a Set using Venn Diagram

The complement of a set using Venn diagram is a subset of U. Let U be the universal set and let A be a set such that A ⊂ U. Then, the complement of A with respect to U is denoted by A' or A\(^{C}\) or U – A or ~ A and is defined the set of all those elements of U which are not in A. Thus, A' = {x ∈ U : x ∉ A}. Clearly, x ∈ A' ⇒ x ∉ A (A – B) is also called the complement … [Read more...]