MTH302 - Counting Principles and Relations

1. Let R = {(1,2), (2,3)} and S = {(2,4), (3,5)}. What is the composition of R and S (R ∘ S)?

a) {(1,4), (2,5)} b) {(1,5), (2,4)} c) {(1,4)} d) {(2,5)}

2. If R is a relation represented by the matrix [1 0; 0 1], what type of relation is R?

a) Reflexive b) Symmetric c) Transitive d) Antisymmetric

3. Which of the following is true for the composition of relations?

a) R ∘ S = S ∘ R b) R ∘ S is always reflexive c) R ∘ S is associative d) R ∘ S is commutative

4. If a relation R is represented by a matrix M, what does a diagonal of 1s in M indicate?

a) R is symmetric b) R is reflexive c) R is transitive d) R is antisymmetric

5. Which of the following is true for the adjacency matrix of a graph representing a relation?

a) It is always symmetric b) It is always reflexive c) It is always transitive d) It depends on the relation

6. If R is a relation represented by a directed graph, what does a loop at a vertex indicate?

a) The relation is symmetric b) The relation is reflexive c) The relation is transitive d) The relation is antisymmetric

7. If R and S are relations, what is the result of R ∪ S?

a) The intersection of R and S b) The union of R and S c) The composition of R and S d) The difference of R and S

8. If a relation R is represented by a matrix M, what does M[i][j] = 1 indicate?

a) i is related to j b) j is related to i c) i and j are not related d) i and j are symmetric

9. Which of the following is true for the composition of two relations R and S?

a) R ∘ S = S ∘ R b) R ∘ S is always reflexive c) R ∘ S is associative d) R ∘ S is commutative

10. If a relation R is represented by a graph, what does the absence of loops indicate?

a) R is reflexive b) R is irreflexive c) R is symmetric d) R is transitive

Relations and Representations