MN01 · Sets, Relations & Functions

5. n(A∩B) = n(A)+n(B)−n(A∪B) = 12+18−25 = 5. Inclusion-exclusion is the most tested set formula in NDA. Memorise it in both directions.

16. Number of subsets = 2ⁿ = 2⁴ = 16. This includes the empty set and the set itself. Number of proper subsets = 2ⁿ−1 = 15.

Set of odd numbers divisible by 2. A number cannot be both odd and divisible by 2 simultaneously — this set has no elements. Option A has 0 elements too (2 is the only even prime; none greater than 2 exist), making it also empty. But "odd numbers divisible by 2" is definitionally impossible, making it the clearest empty set.

{1}. A−B contains elements in A that are NOT in B. From A={1,2,3}: 1 is not in B; 2 and 3 are in B. So A−B={1}. Note B−A={4}, and A−B ≠ B−A in general.

A′ ∩ B′. First De Morgan law: (A∪B)′ = A′∩B′. Second: (A∩B)′ = A′∪B′. These laws "flip" union↔intersection when complement is applied over them.

Reflexive, symmetric, and transitive. All three properties must hold simultaneously for a relation to be an equivalence relation. Missing any one disqualifies it.

(x−3)/2. To find f⁻¹: let y = 2x+3 → x = (y−3)/2. So f⁻¹(x) = (x−3)/2. Verify: f(f⁻¹(x)) = 2·(x−3)/2+3 = x−3+3 = x ✓.

All non-negative reals. x² ≥ 0 for all real x, and equals 0 when x=0. So range = [0, ∞) = non-negative reals. It is NOT all positives because 0 is included.

x. fog(x) = f(g(x)) = f(√x) = (√x)² = x (for x ≥ 0). Note that gof(x) = g(f(x)) = g(x²) = √(x²) = |x|, which is not the same.

20. n(H∪E) = 60+40−20 = 80. Neither = 100−80 = 20. Three-set and two-set Venn diagram counting is a high-frequency NDA topic.

One-one and onto. Bijective = one-one (injective) + onto (surjective). One-one means distinct inputs give distinct outputs; onto means every element of B has a pre-image in A.

4. n(A×B) = n(A)×n(B) → 12 = 3×n(B) → n(B) = 4. The Cartesian product contains all ordered pairs (a, b) with a∈A, b∈B.

{(1,2),(2,1),(2,3),(3,2)}. A relation is symmetric if (a,b)∈R → (b,a)∈R. Here every pair has its reverse also present: (1,2)↔(2,1) and (2,3)↔(3,2). Option A is reflexive, not symmetric (missing cross-pairs).

nᵐ. Each of the m elements in A can be mapped to any of the n elements in B, independently. Total functions = n×n×…×n (m times) = nᵐ.

All reals except ±2. The denominator x²−4=0 when x=±2, making f undefined at these points. Domain = ℝ − {−2, 2}.

30. n(A∪B) = n(A−B)+n(A∩B)+n(B−A) = 15+5+10 = 30. The three parts — A only, both, B only — are disjoint and together make up A∪B.

f(x) = x. The identity function I(x)=x satisfies f∘I = I∘f = f for any function f. It maps every element to itself.

(A∩B) ∪ (A∩C). Distributive law: intersection distributes over union. Similarly, A∪(B∩C) = (A∪B)∩(A∪C).

f(x) = 2x + 1. A strictly increasing (or decreasing) linear function is always one-one. For f(x)=x², f(−2)=f(2)=4 — not one-one. |x| and sin x also fail the horizontal line test.

7. Total subsets = 2³ = 8. Proper subsets exclude the set itself: 8−1 = 7. Some books further exclude the empty set, giving 6 — NDA follows the convention proper subsets = all subsets except the full set, giving 7.