Class 12 Maths - ISC

Relations and Functions

The chapter 'Relations and Functions' in Class 12 ISC Mathematics builds upon foundational set theory by advancing into specialized mappings between sets. Students explore types of relations including reflexive, symmetric, transitive, and equivalence relations, which are crucial for structuring logical arguments. The latter half delves into functions, focusing strictly on injectivity (one-one), surjectivity (onto), and bijectivity, alongside the concept of invertible functions and binary operations. For the ISC board exams, this chapter serves as a high-scoring foundational topic that frequently appears in both short-answer and long-answer sections, testing analytical thinking and rigorous mathematical proof-writing.

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Key Concepts

Reflexive Relation

A relation R on set A is reflexive if every element relates to itself, meaning (a, a) belongs to R for all a in A.

Symmetric Relation

A relation R is symmetric if whenever (a, b) belongs to R, then (b, a) must also belong to R for all a, b in A.

Transitive Relation

A relation R is transitive if whenever (a, b) and (b, c) belong to R, then (a, c) must also belong to R.

Equivalence Relation

A relation is called an equivalence relation if it is simultaneously reflexive, symmetric, and transitive.

One-One (Injective) Function

A function f: A -> B is one-one if distinct elements in domain A have distinct images in codomain B, meaning f(a1) = f(a2) implies a1 = a2.

Onto (Surjective) Function

A function f: A -> B is onto if every element in the codomain B has at least one pre-image in the domain A.

Important Formulas

Total number of relations from set A to set B = 2^(m*n) where n(A)=m and n(B)=n

Total number of reflexive relations on a set with n elements = 2^(n^2 - n)

Total number of equivalence relations cannot be given by a simple formula but can be found using partitions for small sets

Number of one-one functions from A to B (where n(A)=m, n(B)=n) = nPm if m <= n, and 0 if m > n

Number of onto functions from A to B = Summation from k=0 to n of [(-1)^(n-k) * nCk * k^m]

Condition for invertibility: A function is invertible if and only if it is both one-one and onto (bijective)

Board Exam Info

In the ISC Class 12 Mathematics examination, this chapter typically carries around 6 to 8 marks. Questions usually include proving whether a given relation is an equivalence relation, testing functions for injectivity and surjectivity, and finding the inverse of a given bijective function.

Frequently Asked Questions

How do I prove a function is one-one for all real numbers?

Assume f(x1) = f(x2) and solve the equation algebraically. If you can definitively prove that x1 must equal x2 with no alternative solutions, the function is one-one.

What is the easiest way to disprove a relation is transitive?

Provide a specific counterexample. Find elements a, b, and c where (a, b) and (b, c) are in the relation, but (a, c) is missing.

Do I need to find the range to prove a function is onto?

Yes, proving a function is onto requires showing that the range of the function is identically equal to its given codomain.

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