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Sets and Relations

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although, any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

For example: A = {7, 8, 9, 2, 3, 4}.

Let A and B be two non-empty sets. Then, a relation R from A to B is a subset of Cartesian product of the set A and B i.e. (A x B).

Thus, R $\subseteq$ is a relation from A to B. R $\subseteq$ (A x B).

  1. If (a, b) R, then we say that a is related to b and we write a $\in$ b.
  2. If (a, b) R, then we say that a is not related to b and we write a $notin$ b.

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Sets and Relations Formulas

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The Demorgan's Law is the basic and most important formula for sets, which is defined as
(A $\cap$ B) ' = A' U B' and (A U B)' = A' $\cap$ B'

The relation $R \subset A \times A$ is said to be called as
  1. Reflexive Relation: If a R a $\forall$ a $\in$ A.
  2. Symmetric Relation: If aRb, then bRa $\forall$ a, b $\in$ A.
  3. Transitive Relation: If aRb, bRc, then aRc $\forall$ a, b, c $\in$ A.
If any relation R is reflexive, symmetric and transitive in a given set A, then that relation is known as equivalence relation.

Sets and Relations Examples

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Given below are some of the examples on sets and relations.

Solved Examples

Question 1: Write each of the following sets in the tabular form:
  1. A = set of all factors of 24.
  2. B = set of all odd numbers less than 10.
  3. C = set of all prime number less than 30.
  4. D = set of all letters in the word, MATHEMATICS.

Solution:

In the tabular form, we write them as given below.

All factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

Therefore,

A = {1, 2, 3, 4, 6, 8, 12, 24}.

All odd numbers less than 10 are 1, 3,5,7,9.

Therefore,

B = {1, 3, 5, 7, 9}.

All Prime number less than 30 are 2,3,5,7,11,13,17,19,23,29.

Therefore,

C = {2,3,5,7,11,13,17,19,23,29}

The set of all letters in the word MATHEMATICS is given by

Therefore,

D = {M,A,T,H,E,M,A,T,I,C,S}.



Question 2:

Let A = {1, 2, 3} and B = {2,4,6}.

Show that R = {(1, 2), (1, 4), (3, 2), (3, 4)} is a relation from A to B. Find

  1. Domain
  2. Co-Domain
  3. Range

Depict the above relation by an arrow diagram.


Solution:

Here, A = {1,2,3}, B = {2,4,6} and R = {(1,2),(1,4),(3,2),(3,4)}.

Since R (A x B), R is a relation from A to B.

Domain(R) = set of first coordinates of elements of R

= {1,3}.

Range(R) = Set of second coordinates of elements of R

= {2,4}.

We may represent the above relation by an arrow diagram as shown below.

sets and relations

Sets and Relations Practice Problems

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Given below are some of the practice problems on sets and relations.

Practice Problems

Question 1: If A X B = {(1,0), (2,1), (3, -1), (4, 3)}, then find A and B.
Question 2: If R is a relation on N, which is defined as x + 3y = 7, then find the domain of R.
Question 3: If A = {a, b, c, d, e, f } and B = {a, c, j, k, m}, then find
  1. (A $\cap$ B) '
  2. (A U B)'

More topics in Sets and Relations
Relation between Two Sets Properties of Relations
Types of Relations of Sets Equivalence Relation
Ordered Pairs Domain and Range
Cartesian Representation of Relation Arrow Diagram
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