4. Given an example of sets A, B, C and functions f: A → B and g: B → C such that gof: AC is surjective but f is not surjective.

8. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers. a. b.

Prove that if f is a permutation on A, then (f1)1=f.

13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.

Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.

6. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers. a. b.

[Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and multiplication are defined as follows: if and only if and in , Given that is a ring, determine whether is commutative and whether has a unity. Justify your decisions. [Type here]

21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.

Label each of the following statements as either true or false. 9. Composition of mappings is an associative operation.

Label each of the following statements as either true or false. Let f:AB. Then f(A)=B for all nonempty sets A and B.