Math 3312 – Homework Assignment 2

Due February 12, 2004

1. Let A = { 1, 2, 3, 4, 5, 6 }. In each of 1a – 1d, give an example of a relation R on A that has the specified properties, if possible. Explain your answers. (Do not use exactly the same examples as in the text.)

1a. R is both symmetric and antisymmetric.

1b. R is neither symmetric nor antisymmetric.

1c. R is transitive but RÈR^-1 is not.

1d. R is not reflexive nor symmetric nor transitive nor antisymmetric.

2. Let S be a nonempty set and P = { A_i } be a collection of nonempty subsets of S.

Complete the following statement. P is not a partition of S if …

3. See problem 3.59.

Let A be a set of integers, and let ~ be the relation on A×A defined by (a, b) ~(c, d) if and only if a + d = b + c.

3a. Prove ~ is an equivalence relation on A.

3b. Let A = { 1, 2, 3, 4, 5, 6 }. Find the elements of the equivalence class [(3, 5)].

4. See problem 3.60.

Let º be the relation on the set R of real numbers defined by aºb if and only if (b-a)ÎZ, that is, (b-a) is an integer. Prove that º is an equivalence relation on R.