UMASS AMHERST MATH 300, F. HAJIR
HOMEWORK 2: LOGIC AND SETS
1. Reading
You should read all of Chatper 1 in Gilbert/Vanstone as well as Part II of Farshid’s class
notes.
2. Problems from Gilbert/Vanstone Chapter 1
Exercise Set 1 (p. 20): 1-13,25,41-44, 56,58,60, 64, 66,69.
Problem Set 1 (p. 22): 74,76.
3. Problems from Farshid’s brain
1. Prove that P⇒(P∨Q) is a tautology, i.e. its truth table has value “True” in all cases.
2. Consider the following statement.
A: All residents of Amherst MA are residents of Massachusetts.
(a) Rewrite Ain the form of an implication i.e. in the form If P, then Q.
(b) Now give the converse of A.
(c) Is Atrue? Explain.
(d) Is the converse of Atrue? Explain.
3. (a) Suppose the union of ten sets A1∪A2∪ · · · ∪ A10 equals A1. What can you conclude
about these sets?
(b) Suppose the intersection of ten sets B1∩B2∩ · · · ∩ B10 is B1. What can you conclude
about these sets?
4. Consider the sets A={0,1},B={a, b, c}. List the elements of the sets A×A,A×B,
B×A,A×B×A.
5. Prove that if P⇒Qand Q⇒Rand R⇒P, then P, Q, R are all pairwise equivalent.
4. Extra Credit
1. Construct a sequence of sets S1, S2, S3, ... (one for each natural number) such that for
any finite subset {i1, . . . , in} ⊂ Nof the natural numbers, the intersection Si1∩ · · · ∩ Sinis
an infinite set, but ∩n≥1Sn={}.
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