Sample Solution
a.
AAB
B AB
2A 2AB
2B 2AB
2A 2B 2AB
Every set of elements from A (or from B) is a set of elements from A B.
If A B then A B = B and 2AB = 2B 2A 2B , so the two sets are equal. The result
also holds if B A.
If A 6 B and B 6 A, then 2A 2B
Homework 2
CS3378/CS5338
Fall 2003 C. Hazlewood
1. In general, a relation on a set can be reflexive or not reflexive, symmetric or not symmetric, and transitive or not transitive eight possible combinations in all. Construct
relations on the set cfw_1, 2,
1.3.5 Let f : A B. Show that the following relation R is an equivalence relation on A: (a, b) R
if and only if f (a) = f (b).
Sample Solution: Show that R is reflexive, symmetric, and transitive.
Let a A. Since f (a) = f (a) (by the reflexive property of
1.7.2 Prove the following.
a. (wR )R = w.
Proof by induction on the length of w.
Base case: |w| = 0. w = . (wR )R = (R )R = ()R = = w.
Induction hypothesis: if 0 |x| n, then (xR )R = x.
Induction step: Suppose |w| = n + 1. Then we can write w = xa, where
Homework 1
CS3378/CS5338
Fall 2003 C. Hazlewood
1. Suppose that A and B are sets. What is the relationship between 2AB and 2A 2B ?
(Under what circumstances are they equal? If they are not equal, is one necessarily a
subset of the other, and if so, which