MATH 135 Test 1 Sample Test
This is based on a test from 2019, with some question substitutions.
Suggested time: 90 minutes
Question 1
For each part, full marks will be given for a correct answer which is placed in
the box provided. If the final answer in the box is incorrect, part marks may
be awarded based on work shown.
[1]
(a) Determine a value of x such that (2 < x < 18) ∧ (2 | x) ∧ (x | 18).
[2]

 

5
5
X
X
(b) Evaluate 
j2 − 
j .
j=3
j=3
[1]
(c) Given an expression for the coefficient of x2 in the expansion of (1 − 6x)8 .
[2]
(d) Express the following statement symbolically. Do not use any words.
There exists an integer that is both a perfect square and a perfect cube.
[1]
(e) Determine a value of x and a value of y that disproves the following:
∀x ∈ Z, ∀y ∈ Z, (x2 + y 2 = 1) =⇒ (x = 1 ∨ x = −1).
x=
and y =
1
Question 2
For each part, circle True or False. No justification is needed.
Given a variable x, let P (x) denote the open sentence x ≥ 0, and let Q(x) denote the open
sentence x < 0.
[1]
[1]
[1]
[1]
[1]
[1]
(a) P (0) ∧ Q(0)
True
False
(b) ∀x ∈ R, P (x) ⇒ ¬Q(x)
True
False
(c) (∀x ∈ R, P (x)) ∨ (∀x ∈ R, Q(x))
True
False
(d) ∀x ∈ R, (P (x) ∨ Q(x))
True
False
(e) (∀x ∈ N, P (x)) ∨ (∀x ∈ N, Q(x))
True
False
(f) ∀x ∈ N, (P (x) ∨ Q(x))
True
False
Question 3
Let A and B be statement variables.
[3] (a) Complete the following truth table.
A
T
T
F
F
[1]
B
T
F
T
F
¬B
A ∧ (¬B)
B⇔A
(A ∧ (¬B)) ∨ (B ⇔ A)
B⇒A
(b) Is (A ∧ (¬B)) ∨ (B ⇔ A) logically equivalent to B ⇒ A?
Circle one:
YES
NO
Question 4
[5] Let a, b ∈ Z with a ≥ 2. Prove that if a 6= 13, then a - (3b + 1) or 3a - (7b − 2).
2
Question 5
Let x, y ∈ R. Consider the implication S:
If xy > 6, then x > 2 and y > 3.
[1]
[1]
[1]
[1]
[1]
[2]
(a) State the hypothesis of S.
(b) State the conclusion of S.
(c) State the converse of S.
(d) State the contrapositive of S.
(e) State the negation of S in a form that does not contain an implication.
(f) Indicate clearly whether the given implication S is true or false for all x, y ∈ R. Then
prove or disprove the statement.
Circle the correct answer: True
False
Question 6
√
√ aj
[5] A sequence is defined by a1 = 2 and aj+1 =
for all j ∈ N.
2
Prove that, for every n ∈ N, an < 2.
Question 7
[5] Let a, b be non-negative integers. Prove that 3a = 8b if and only if a = b = 0.
Question 8
For each of the following statements indicate clearly whether the statement is true or false
and then prove or disprove the statement.
[2] (a) ∀x ∈ R, ∃y ∈ R, x + 2y = 0.
Circle the correct answer: True
False
[3] (b) ∃y ∈ R, ∀x ∈ R, x + 2y = 0.
Circle the correct answer: True
False
3