Mat 300, Spielberg Test 1 Review Problems Spring 07
1. (i) Show that the formulas (p∨q)→rand (p→r)∧(q→r) are logically equivalent.
Do this by using known equivalences (such as commutativity, associativity, distributivity,
and DeMorgan’s laws) to modify one of the formulas until it becomes the other. Show all
steps in the process.
(ii) Do the same for p→(q→r) and (p∧q)→r.
2a. Rewrite the following statements symbolically without using union, intersection, con-
tainment, or negation symbols (but you may use logical symbols in this problem):
x∈\
i∈I
Ai
x6∈ \
i∈I
Ai
2b. Rewrite the following statements in equivalent form using only the symbols: A,B,C,
∪,∩,⊆, (, ), =, 6=, ∅. Show all intermediate steps.
A\B⊆C
A\B6⊆ C
3. Prove that A\Ti∈IBi=Si∈I(A\Bi).
4. Let A,B, and Cbe sets. Suppose that A⊆B0∪C. Prove that A∩B⊆C.
5. Let {Ai:i∈I}and {Bj:j∈J}be indexed families of sets. Suppose that for all i∈I
and for all j∈Jwe have Ai⊆Bj. Prove that
[
i∈I
Ai⊆\
j∈J
Bj.
6. Use axioms (A11) and (A12) to prove that the multiplicative identity in Ris unique.
7. Use axioms (A10), (A11), (A12) and (A13) to prove that the multiplicative inverse of
a non-zero real number is unique.
8. Use properties 9b and 9h on page 59 to prove that for any real numbers aand b, if
a < b < 0 then a2> b2.
9. Suppose that |x+ 10|<5. Prove that |x|>5.
10. (i) Prove that for every natural number n,
1·2+2·3+3·4 + · · · +n(n+ 1) = 1
3n(n+ 1)(n+ 2).
(ii) Prove that for every natural number n≥4,
20n < 3n.
11. Problems 1 and 2 on the sheet Homework Problems C, and Quiz 4.