Relations and Functions
Punjab Board · Class 12 · Mathematics
Flashcards for Relations and Functions — Punjab Board Class 12 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.
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Explore the full setDetermine if relation R = {(a,b) : a - b = 10} in set A = {1,2,3,4} is reflexive, symmetric, or transitive.
Answer
Step 1: Check reflexivity - Does (a,a) ∈ R for all a? Test (1,1): 1-1=0≠10. So NOT reflexive. Step 2: Check symmetry - If (a,b) ∈ R, is (b,a) ∈ R? Since no pairs exist in R (no two elements differ by…
Show that R = {(a,b) : 2 divides (a-b)} in set Z is an equivalence relation.
Answer
Step 1: Reflexivity - Does 2 divide (a-a) = 0? YES, 0 is divisible by 2. ✓ Step 2: Symmetry - If (a,b) ∈ R, then 2|(a-b). Does 2|(b-a)? Since (b-a) = -(a-b) and 2 divides (a-b), it divides -(a-b). So…
Verify if relation R = {(L₁, L₂) : L₁ is perpendicular to L₂} in set L of all lines is reflexive, symmetric, or transitive.
Answer
Step 1: Reflexivity - Can a line be perpendicular to itself? NO. (L₁, L₁) ∉ R. NOT reflexive. Step 2: Symmetry - If L₁ ⊥ L₂, is L₂ ⊥ L₁? YES, perpendicularity is mutual. (L₁, L₂) ∈ R ⟹ (L₂, L₁) ∈ R.
For relation R = {(a,b) : both a and b are odd or both are even} in A = {1,2,3,4,5,6,7}, identify the equivalence classes.
Answer
Step 1: Since R is an equivalence relation (can verify: reflexive, symmetric, transitive), it partitions A into equivalence classes. Step 2: Elements related to 1 (odd) - All odd numbers in A: [1] = …
Check if function f: ℕ → ℕ defined by f(x) = 2x is one-one and onto.
Answer
Step 1: Check one-one (injective) - If f(x₁) = f(x₂), does x₁ = x₂? Assume 2x₁ = 2x₂ ⟹ x₁ = x₂. YES, one-one. ✓ Step 2: Check onto (surjective) - For every y ∈ ℕ, does there exist x ∈ ℕ such that f(x…
Prove that f: ℝ → ℝ given by f(x) = 2x is one-one and onto.
Answer
Step 1: Check one-one - If f(x₁) = f(x₂), then 2x₁ = 2x₂ ⟹ x₁ = x₂. YES, one-one. ✓ Step 2: Check onto - For any y ∈ ℝ, find x ∈ ℝ such that f(x) = y. Set 2x = y ⟹ x = y/2. Since y/2 ∈ ℝ for all y ∈ …
Determine if f: ℝ → ℝ given by f(x) = x² is one-one and onto.
Answer
Step 1: Check one-one - If f(x₁) = f(x₂), does x₁ = x₂? Test: f(-1) = 1 and f(1) = 1, but -1 ≠ 1. NOT one-one. ✗ Step 2: Check onto - For every y ∈ ℝ, does there exist x ∈ ℝ such that f(x) = x² = y?
Find gof and fog if f(x) = cos x and g(x) = 3x² for f,g: ℝ → ℝ. Are they equal?
Answer
Step 1: Find gof(x) = g(f(x)) = g(cos x) = 3(cos x)² = 3cos²x Step 2: Find fog(x) = f(g(x)) = f(3x²) = cos(3x²) Step 3: Check equality - Are 3cos²x and cos(3x²) equal? Test x = 0: - gof(0) = 3cos²(0…
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