MAT 300 Mathematical Structures / 100
Test 1 February 20, 2004 name
1. Use truth tables to decide whether the statements
(A⇒B)⇒Aand (B⇒A)⇒Aare equivalent.
1 2 3 4 5 6
10 10 20 30 15 15
2. Fill in the empty boxes with one of ⇒,⇐,⇔, or N (neither).
xis a real number and x < 10 xis a real number and x≤10
xis a real number and x < 12 xis a real number and x < 10
xis a natural number divisible by 10 xis a natural number divisible by 20
xis a real number and x26= 0 xis a real number and x6= 0
xis a real number and x2<0xis a real number and x < 0
3. Let L(x, y) stand for “xloves y.”
a. Use quantifiers and this predicate to write the following in symbolic form
(i) “Somebody loves everybody.” (ii) “Somebody loves only her/himself.”
b. In plain English negate the sentences (i) and (ii). Use positive language as much as possible.
c. Use quantifiers and predicates to express the negated sentences.
Simplify each expression so that no quantifier or implication remains negated.
4.a. Consider the following sentences (some may have the same meanings) and fill in all empty boxes
with Y (yes), N (no), T (true), F (false), n/a (not applicable). Consider only real numbers.♣
predicate statement true/false
A = “x2+ 2x+ 101 = 0 has a solution.”
B = “There exists xsuch that x2+ 2x+ 101 = 0.”
C = “x < 0”
D = “x2+ 2x+ 101 = 0.”
E = “x2+ 2x+ 101 = 0 has a negative solution.”
F = “There exists x < 0such that x2+ 2x+ 101 = 0.”
G=“If x2+ 2x+ 101 = 0 then x < 0.”
H = “Every solution xof x2+ 2x+ 101 = 0 is negative.”
b. State the converse of G– use positive terms as much as possible.
c. State the contrapositive of G– use positive terms as much as possible.
d. State the negation of G– use positive terms as much as possible.
5.a. State one of the deMorgan’s laws.
b. State the standard definitions for “is a subset of”, and for “power-set”.
c. Suppose that Aand Bare sets. Prove: “If A⊆Bthen P(A)⊆ P (B).”
6.a. State the axiom of induction.
b. Use induction to prove: “If n∈Z+then
n
P
k=1
(2k−1) = n2.”