Relations — Important Questions

Reflexive, Symmetric, Transitive

Step 1: Check reflexive: (1,1), (2,2), (3,3) ∈ R, so reflexive. Step 2: Check symmetric: For each (a,b) ∈ R, (b,a) ∈ R (trivially true as all pairs are of form (a,a)). Step 3: Check transitive: For (a,b) and (b,c) in R, (a,c) ∈ R (trivially satisfied). This is the identity relation on A.

Yes

Step 1: Check transitive property: if (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R. Step 2: We have (1,2) ∈ R and (2,3) ∈ R. Step 3: Check if (1,3) ∈ R. Yes, (1,3) ∈ R. Step 4: No other pairs in R form a chain, so R is transitive.

Domain = {2, 3, 4, 5}, Range = {4, 6, 8, 10}

Step 1: Domain = set of all first elements of ordered pairs. Step 2: Domain = {2, 3, 4, 5}. Step 3: Range = set of all second elements of ordered pairs. Step 4: Range = {4, 6, 8, 10}.

64

Step 1: Number of relations from A to B = 2^(n(A) × n(B)). Step 2: n(A × B) = 2 × 3 = 6. Step 3: Number of relations = 2^6 = 64. Step 4: Each relation is a subset of A × B, and a set with 6 elements has 2^6 subsets.