Relations And Functions

Concept: A relation is an equivalence relation if it satisfies three properties: \begin{itemize} \item Reflexive \item Symmetric \item Transitive \end{itemize} Given relation:

(a%2C%20b)%5C%2C%20R%5C%2C%20(c%2C%20d)%20%5Ciff%20a%20%2B%20d%20%3D%20b%20%2B%20c

Step 1: Reflexive property.
A relation is reflexive if every element is related to itself. Check:

(a%2C%20b)%5C%2C%20R%5C%2C%20(a%2C%20b)

Condition:

a%20%2B%20b%20%3D%20b%20%2B%20a

This is true for all natural numbers. Hence,

R

is reflexive.
Step 2: Symmetric property.
A relation is symmetric if:

(a%2C%20b)%5C%2C%20R%5C%2C%20(c%2C%20d)%20%5CRightarrow%20(c%2C%20d)%5C%2C%20R%5C%2C%20(a%2C%20b)

Given:

a%20%2B%20d%20%3D%20b%20%2B%20c

Rewriting:

c%20%2B%20b%20%3D%20d%20%2B%20a%0A%5CRightarrow%20c%20%2B%20b%20%3D%20d%20%2B%20a

Which is equivalent to:

c%20%2B%20b%20%3D%20d%20%2B%20a%0A%5CRightarrow%20(c%2C%20d)%5C%2C%20R%5C%2C%20(a%2C%20b)

Hence,

R

is symmetric.
Step 3: Transitive property.
A relation is transitive if:

(a%2C%20b)%5C%2C%20R%5C%2C%20(c%2C%20d)%20%5Ctext%7B%20and%20%7D%20(c%2C%20d)%5C%2C%20R%5C%2C%20(e%2C%20f)%0A%5CRightarrow%20(a%2C%20b)%5C%2C%20R%5C%2C%20(e%2C%20f)

Given:

a%20%2B%20d%20%3D%20b%20%2B%20c%20%5Cquad%20%5Ccdots%20(1)

c%20%2B%20f%20%3D%20d%20%2B%20e%20%5Cquad%20%5Ccdots%20(2)

Add (1) and (2):

a%20%2B%20d%20%2B%20c%20%2B%20f%20%3D%20b%20%2B%20c%20%2B%20d%20%2B%20e

Cancel common terms

c%20%2B%20d

a%20%2B%20f%20%3D%20b%20%2B%20e

Thus,

(a%2C%20b)%5C%2C%20R%5C%2C%20(e%2C%20f)

Hence,

R

is transitive. Conclusion:
Since the relation is reflexive, symmetric, and transitive,

R

is an equivalence relation.

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Relations And Functions

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs