Math 266 - Final Exam Practice Questions
1. Let pbe a prime and a∈Z. Here is a logical statement:
“If pdivides a2, then pdivides a.”
What is the contrapositive statement?
2. Let Pand Qbe propositions. Find a statement logically equivalent to P∨Qusing only
negation (∼) and conjuction (∧) operations.
3. Prove that there is a rational number between any two unequal real numbers.
4. Define, for x, y ∈R:
max(x, y) = (xif x≥y;
yif y > x.
Prove, for any z∈R, that if z≥xand z≥y, then z≥max(x, y).
5. For x, y ∈R, give a precise definition of min(x, y), the minimum of xand y. Prove that
min(x, y) = −max(−x, −y).
6. Let {An}∞
n=1 and {Bn}∞
n=1 be two families of sets. Prove
∞
\
n=1
An!∪ ∞
\
n=1
Bn!⊆
∞
\
n=1
An∪Bn.
7. Let Rbe a relation on a set A, and define a new relation Son Aby xSy iff xRy or yRx.
Prove that Sis symmetric.
8. Define a relation on Z7(the integers modulo 7) by xRy if x≡y+1 (mod 7) or x≡y+ 2
(mod 7). Draw the digraph for R.
9. For a, m ∈N, define f:Zm→Zmby f(x) = ax. Prove fis a bijection if and only if
GCD(m, a) = 1.
10. Let f:Q∩(0,∞)→N×Nby f(p/q)=(p, q), where we assume p/q is a rational number
in lowest terms.
(a) Is fone-to-one? Prove or disprove.
(b) Is fonto? Prove or disprove.
11. Let f:R→Rby f(x) = sin(x).
(a) Find f([−14.000001,182.632]).
(b) Find f−1([−14.000001,182.632]).
12. A function f:R→Ris called even if f(−x) = f(x) for all x∈R.
(a) Give three examples of even functions.
(b) Prove: For any function g:R→R, the function f(x) = g(x2) is even.
