Homework 1
CSC2700-001
Section 1.1:
2:
a) imperative
b) declarative statement that is FALSE
c) declarative
d) interrogative
e) exclamatory
f) declarative statement that is TRUE
g) declarative
4:
a) 5, -1
b) 2
c) every non-zero integer
d) +/-1 & +/-2
e) 0
f) 0, 2
6:
a) 1
b) -2
8:
a) x=1/2, -2 & y=1
Section 1.2:
2:
a) √3 ≤ 1.7
b) The integer 0 IS NOT even
c) The number 7 IS a root of the equation
6:
a) 1
b) -1
c) 4
12:
a) x≥-3 and x≤3
b) The integer a is NOT odd or the integer b is NOT even.
14:
a)
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
~Q
F
T
F
T
PvQ
T
T
T
F
~(PvQ)
F
F
F
T
(~P)v(~Q)
F
T
T
T
b)
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
~Q
F
T
F
T
P^Q
T
F
F
F
~(P^Q)
F
T
T
T
20: FALSE
P
Q
R
QꚚR
PvQ
PvR
Pv(QꚚR)
(PvQ)Ꚛ(PvR)
T
T
T
F
T
T
T
F
T
T
F
T
T
T
T
F
T
F
T
T
T
T
T
F
T
F
F
F
T
T
T
F
F
T
T
F
T
T
F
F
F
T
F
T
T
F
T
T
F
F
T
T
F
T
T
T
F
F
F
F
F
F
F
F
Section 1.3:
4:
1) True
2) False
3) True
10:
a) 7 is an even integer, then 0 is a positive integer : TRUE
b) 0 is a positive integer, then 7 is an even integer: TRUE
12:
a) If 110 is even, then 101 is even: TRUE
b) If 101 is even, then 110 is even: TRUE
c) If 110 is not even, then 101 is not even: FALSE
16:
P
Q
~P
Q ⇒ (∼ P)
P ∧ (Q ⇒ (∼ P))
T
T
F
F
F
T
F
F
T
T
F
T
T
T
T
F
F
T
T
T
20:
a) If today is Saturday or Sunday, then I do not have class today
b) If I have class today, then today is not Saturday or Sunday
c) Today is not Saturday or Sunday or I do not have class today
d) If I do not have class today, then today is Saturday or Sunday
e) I have class today or today is Saturday or Sunday
f) If today is not Saturday or Sunday, then I have class today
24:
P
T
a) TRUE
Q
T
R
T
P⇒Q
T
P⇒R
T
Q^R
T
(P ⇒ Q) ∧ (P ⇒ R)
T
P ⇒ (Q ∧ R)
T
(~P)^(~Q)
F
F
F
T
T
T
T
F
F
F
F
P
T
T
T
T
F
F
F
F
P
T
T
T
T
F
F
F
F
P
T
T
T
T
F
F
F
F
P
T
T
T
T
F
T
F
F
T
T
F
F
b) FALSE
Q
T
T
F
F
T
T
F
F
c) TRUE
Q
T
T
F
F
T
T
F
F
d) TRUE
Q
T
T
F
F
T
T
F
F
e) FALSE
Q
T
T
F
F
T
F
T
F
T
F
T
F
T
F
F
T
T
T
T
F
T
F
T
T
T
T
F
F
F
T
F
F
F
F
F
F
T
T
T
T
F
F
F
T
T
T
T
R
T
F
T
F
T
F
T
F
P⇒R
T
F
T
F
T
T
T
T
Q⇒ R
T
F
T
T
T
F
T
T
PvQ
T
T
T
T
T
F
T
F
(P ⇒ R) ∧ (Q ⇒ R)
T
F
T
F
T
F
T
T
(PvQ) ⇒ R
T
F
T
F
T
T
T
T
R
T
F
T
F
T
F
T
F
P⇒Q
T
T
F
F
T
T
T
T
P⇒R
T
F
T
F
T
T
T
T
QvR
T
T
T
F
T
T
T
F
(P ⇒ Q) v (P ⇒ R)
T
T
T
F
T
T
T
T
P ⇒ (Q v R)
T
T
T
F
T
T
T
T
R
T
F
T
F
T
F
T
F
P⇒R
T
F
T
F
T
T
T
T
Q⇒ R
T
F
T
T
T
F
T
T
R
T
F
T
F
T
P⇒Q
T
T
F
F
T
Q⇒ R
T
F
T
T
T
P^Q
T
T
F
F
F
F
F
F
(P ⇒ Q) v (Q ⇒ R)
T
F
T
T
T
T
T
T
(P ⇒ Q) ^ (Q ⇒ R)
T
F
F
F
T
P⇒R
T
F
T
F
T
(P^Q) ⇒ R
T
F
T
T
T
T
T
T
F
F
F
T
F
F
F
T
F
T
T
T
F
T
T
F
T
T
T
T
T
Section 1.4:
2:
IF (-1)2 , then TRUE because both statements are true. IF –(1)2, then FALSE because -1 is NOT
greater than 0
6:
5n+7 is even if and only if n is odd
8:
a) a=2 b=4
b) a=1 b=4
12:
a) TRUE
P
Q
~P
~Q
P⇔Q
(∼ P) ⇔ (∼ Q)
T
T
F
F
T
T
T
F
F
T
F
F
F
T
T
F
F
F
F
F
T
T
T
T
b) TRUE
P
Q
~P
~Q
(P ∧ Q)
(∼ P) ∧ (∼
P⇔Q
(P ∧ Q) ∨ ((∼ P) ∧
Q)
(∼ Q))
T
T
F
F
T
F
T
T
T
F
F
T
F
F
F
F
F
T
T
F
F
F
F
F
F
F
T
T
F
T
T
T
c) TRUE
P
Q
~Q
P⇔Q
~( P ⇔ Q)
P ⇔ (∼ Q)
T
T
F
T
F
F
T
F
T
F
T
T
F
T
F
F
T
T
F
F
T
T
F
F
Section 1.5:
4:
a) TRUE
P
T
T
F
F
b) TRUE
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
P⇒Q
T
F
T
T
(∼ P) ⇒ (P ⇒ Q)
T
T
T
T
Q
T
F
T
F
P∧Q
T
F
F
F
P⇒Q
T
F
T
T
(P ∧ Q) ⇒ (P ⇒ Q)
T
T
T
T
8:
TAUTOLOGY
P
Q
T
T
F
F
T
F
T
F
~Q
P ∧ (∼ Q)
P∨Q
F
T
F
T
F
T
F
F
T
T
T
F
(P ∧ (∼ Q)) ⇒ (P ∨
Q)
T
T
T
T
Section 2.1:
4:
a) x is an element of the set of real numbers such that it satisfies the equation x5 − 3x3 − x2 +
4x − 1 = 0
b) n is an element of the set of integers such that it remains an integer when divided by 3
c) The set of all odd integers
6:
a) yes
b) no
c) no
d) no
8:
a) 1
b) 2
c) 3
14:
C
16:
A={1, 2, 3} B={1, 2, 3, 4} C={1, 2, 3, 4}
20:
a) P(A)={∅}
b) P(B)={0,{ ∅}}
c) P(C)={ ∅, {0}}
d) P(D)={ ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
24:
2048
Section 2.2:
4:
a) {1, 2}
b) {3}
c)
8:
ALL sets are equal
D
12:
a) 𝑆 ∪ 𝑇
b) T – S
c) S – T
d) S ⊕ T
e) S ∩ T
16:
Section 2.3:
2:
A2 = {(0, 0), (0, {1}), ({1}, 0), ({1}, {1})}
4:
𝐵𝑥𝐵 = {(1, 3), (2, 3), (3, 1), (3, 2), (3, 3)}
6:
P(A)xP(B) = {(∅, ∅), (∅, {∅}), ({1}, ∅), ({1}, {∅}), ({2}, ∅), ({2}, {∅}), (A, ∅), (A, {∅})}
8:
(A × B) ∩ P(A × B) = ∅
Section 2.4:
2:
{1, 2, 3}
6:
a) {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}, {13, 14, 15}}
b) {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}
8:
{∅, {1}, {2}, {1, 2}}