1. Prove that A ∪ B = ? and A∩ B = ∅<=> B = A' where ? is the universal set.

2. Let A?B = { x : x ∈ A ∪ B and x ∉ A ∩ B } for any finite sets A and B. Show that A? (A ∩ B) = A - B.

3. If A ⊆ B, then which of the followings is true and why?
a) A' = B
b) A - B = ∅
c) A'⊆ B
d) A ∩ B ≠∅
e) None of the above

4. What is the necessary and sufficient condition for A - B = A? Why?
a) A = B
b) A ⊆ B
c) B ⊆ A
d) B = ∅
e) None of the above

5. Let B = {{ ∅ }, ∅, 0, (0, ∅)}. Which of the followings is not true? Why?
a) { ∅ } ⊆ B
b) { ∅, 0 } ⊆ B
c) ∅⊆ B
d) { { ∅ } } ⊆ B
e) None of the above

6. Show by truth table that if p ⇒ q and r'⇒ q. is always true, then r'⇒ p' is always true.

7. Prove without using a truth table: [(p ⇒ q) ∧ (p ⇒ r)] ≡ [p ⇒ (q ∧ r)].

8. Prove the following statement: "If n is an integer, then n2has remainder 0 or 1 upon dividing by 3".

9. Write the negation (in words) of the following claim: "If Jack and Jill climb up the hill, then they fall down and like pails of water".

10. Definition: If R ⊆A x B is a relation, then its inverse R-1⊆ B x A is the set R-1 = {(b, a) ∈B x A : (a, b) ∈ R}.

Let R ⊆ {1, 2, 3, 4} x {1, 2, 3, 4}be a relationR = {(1, 3), (1, 4), (2, 2), (2, 4), (3, 1), (3, 2), (4, 4)}.

a) Compute R-1.

b) Compute the relations R ∪R-1 and R∩ R-1, and check if they are

I. Symmetric
II. Reflexive
III. Asymmetric
IV. Antisymmetric
V. Complete
VI. Transitive