# Math 366 Practice Problems for Exam 2

1.Constructive Proof of an Existential Statement.

(a) Prove that there is an even integer n such that n mod 3 = 1.

(b) Prove that there exists a rational number q such that 9 q^{2} = 4.

(c) Prove that there exist two real numbers whose product is less than their
sum .

(d) Prove that there exist two real numbers which are not equal to each other
and whose product

is equal to their sum .

(e) Prove that there is an odd integer n such that n > 1 and n has the form 3k +
1 for some

integer k.

2. Direct Proof of a Universal Statement.

(a) Prove that if n is an integer which is divisible by 6 then n is divisible by
3.

(b) Prove that for any integers a, b, c, and d, if a divides b and c divides d
then a · c divides b · d.

(c) Prove that the product of two odd integers is odd.

(d) Prove that for any sets A, B, and C, if A B and A
C then A
B ∩ C.

(e) Prove that if n is an integer which is divisible by 5 then 3n is divisible
by 15.

(f) Prove that for any rational numbers a and b, if a ≠ 0 then there is a
rational number x such

that ax + b = 0.

(g) Show that the reciprocal of any nonzero rational number is rational.

3. Proof by Cases.

(a) Prove that for every integer n, n and n + 2 have the same parity (i.e.
either n and n + 2 are

both even or n and n + 2 are both odd).

(b) Prove that for any integer n, n^{2} + n is even.

(c) Prove that for any integers n and m, n^{2}+3m ≠ 2. Hint: Use an argument by
cases depending

on what the remainder is when n is divided by 3.

(d) Prove that for any sets A, B, and C, if A
C then A ∪ (B ∩ C)
C.

4. Mathematical Induction .

(a) Prove that for any integer n, if n ≥ 0 then 4 divides 5^{n} -
1.

(b) Prove that for any integer n, if n ≥ 1 then 4 divides 6^{n} - 2^{n}.

(c) Show that 2n + 1 < 2^{n} for every integer n with n ≥ 3.

(d) Using the fact that 2n+1 < 2^{n} for every integer n with n ≥ 3, show that for
every integer n,

if n ≥ 5 then n^{2} < 2^{n}.

5. Strong Mathematical Induction and the Well- Ordering Principle .

(a) Suppose is a sequence defined as follows:

for all integers k ≥ 2.

Prove that for all integers n ≥ 0.

(b) Suppose is a sequence defined as follows:

for all integers k ≥ 2.

Prove that for all integers n ≥ 0.

6. Proofs by Contradiction and Contraposition.

(a) Prove that there is no smallest real number x such
that 1 < x < 2..

(b) For any integer n, if n^{2} is not divisible by 3 then n is not
divisible by 3.

7. Computations with Sets.

(a) Let A = {a, c, d} and B = {b, c, f} be subsets of the universal set U = {a,
b, c, d, e, f, g}.

Compute A ∪ B, A ∩ B, A − B, A^{c}, and A × B using “bracket” notation.

(b) Let A = {1, 3} and B = {2, 3} be subsets of the universal set U = {0, 1, 2,
3, 4}. Compute

A ∪ B, A ∩ B, A − B, A^{c}, and A × B using “bracket” notation.

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