Relations and Functions – important questions for Boards 2027
CBSE Class 12 Maths
For A = {1, 2}, let R = ∅. How many ordered pairs in A are related under R?
Answer
∴ Number of ordered pairs related under R = 0
Let A = {1, 2} and R = {(1, 1), (1, 2)}. Write A × A and identify the empty and universal relations in A. Verify that R is a subset of A × A.
Answer
A × A = {(1, 1), (1, 2), (2, 1), (2, 2)} Empty relation = ∅ Universal relation = A × A = {(1, 1), (1, 2), (2, 1), (2, 2)} R = {(1, 1), (1, 2)} ∴ R ⊂ A × A
Which of the following conditions makes a function f : X → Y bijective? (A) f is one-one only (B) f is onto only (C) f is both one-one and onto (D) f is neither one-one nor onto
Answer
(C) f is both one-one and onto — a function is bijective if it is both one-one and onto.
On A = {1, 2, 3}, let R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. Check whether R is reflexive, symmetric and transitive, giving a reason for each conclusion.
Answer
Reflexive: (1, 1), (2, 2), (3, 3) ∈ R ∴ R is reflexive. Symmetric: (1, 2) ∈ R and (2, 1) ∈ R; the diagonal pairs are symmetric as well. ∴ R is symmetric. The only non-diagonal pairs are (1, 2) and (2, 1); their compositions yield only (1, 1) and (2, 2), which belong to R. Compositions involving diagonal pairs also yield pairs in R. ∴ R is transitive.
Let f : A → B, where A = {1, 2, 3} and B = {2, 4, 6}, be defined by f(x) = 2x. Find the range of f and use it to check whether f is onto. State the relationship between the range and co-domain when a function is onto.
Answer
A function is onto if and only if its range equals its co-domain. f(1) = 2(1) = 2 f(2) = 2(2) = 4 f(3) = 2(3) = 6 Range(f) = {2, 4, 6} = B ∴ f is onto.
An equivalence relation on A = {1, 2, 3, 4} has classes {1, 3} and {2, 4}. Find the equivalence class containing 3.
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On A = {1, 2, 3}, let R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. Given that R is reflexive, symmetric and transitive, determine whether R is an equivalence relation, with a reason.
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Let f be a function from a three-element set to itself. If f is onto, which statement must be true? (A) f is many-one (B) f is one-one (C) f is not onto (D) f has exactly two images
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On Z, define a relation R by a R b if and only if 2 divides a − b. Show that R has exactly two equivalence classes and write them explicitly.
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For f : R → R defined by f(x) = 2x, prove that f is one-one and onto.
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21 more probable questions in this chapter, every one with a full answer.
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