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.

About Relations And Functions - ASSAM-BOARD-CLASS-12

Relations And Functions is a vital chapter for ASSAM-BOARD-CLASS-12 aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Relations And Functions PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Relations And Functions carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

About Relations And Functions - ASSAM-BOARD-CLASS-12

Frequently Asked Questions

Why focus on Relations And Functions PYQs?

How to best use this analysis?

Relations And Functions

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

About Relations And Functions - ASSAM-BOARD-CLASS-12

Frequently Asked Questions

Why focus on Relations And Functions PYQs?

How to best use this analysis?

Chapter Questions
1 MCQs