MAT 300, Spielberg
Midterm Solutions
Spring 2013
1. Consider the following statement: Someone in the Trojan army ran away from the battle. Let T (x)
denote the sentence x is in the Trojan army, and let B (x) denote the sentence x fought in the
battle, whe

MAT 300, Spielberg
Midterm Practice
Spring 2013
All proofs must be written with complete sentences in correct English. Be sure to include all steps in your
proofs. Do not use logical symbols in your proofs.
ALWAYS SHOW YOUR WORK! (The answer, alone, is no

MAT 300
QUIZ 1 Solutions
1/16/13, Spielberg
1. Use the laws of logical equivalence to simplify the following formula. Use only one law at
a time. Your answer should be as simple as possible.
(P Q) (P Q).
Solution.
(P Q) (P Q)
(P Q) (P Q)
(Q P ) (P Q)
Q (P

MAT 300
QUIZ 2 Solutions
1/23/13
1. Verify the identity (A B ) \ C = (A \ C ) (B \ C ) by writing out (using logical
symbols) what it means for an object x to be an element of each side, and then using logical
equivalences. Show all steps of the vericatio

MAT 300, Spielberg
QUIZ 4 Solutions
Fall, 2013
Analyze the logical forms of the following statements. You may use the symbols , , =,
=, , , , , , and , but not , , P , , , \, cfw_, , or . Your answers should be as
simple as possible given this constraint.

MAT 300, Spielberg
QUIZ 5 Solutions
Spring 2013
Prove that for every real number x, if |2x 6| > x then |x 4| > 2.
Proof 1. Let x R. Suppose that |2x 6| > x. Then either 2x 6 > x, or 2x 6 < x.
In the rst case we suppose that 2x 6 > x. Then x > 6. Therefore

MAT 300, Spielberg
QUIZ 6 Solutions
Spring, 2013
Let A, B , C and D be sets. Suppose that B and D are not disjoint. Prove that if A B and
C D are disjoint, then A and C are disjoint. (Your proof should be written in complete
sentences, and should not cont

MAT 300, Spielberg
QUIZ 9 Solutions
Spring 2013
1. Let g : A B . Give the precise denition: g is one-to-one if . . .
Solution. g is one-to-one if for all x1 , x2 A, if g (x1 ) = g (x2 ) then x1 = x2 .
2. Let f = cfw_(x, n) R Z | n x < n + 1. Then f : R Z

MAT 300
QUIZ 11 Solution
Spielberg, S13
Use mathematical induction to prove that for all natural numbers n,
1
0 + 1 + 2 + + n = n(n + 1).
2
1
Proof. For the base case we have 0 = 0(0 + 1). For the inductive step, let n N. Suppose
2
1
that 0 + 1 + 2 + + n