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b) What time is it? Safiirka Islaamka 10 Xaqiiqo Oo La Yaab Leh Oo Ku Saabsan Shukaansiga Sheikh Axmed... (1) \( +\ldots-\ldots \) e) The moon is made of green cheese. 7) Let \( p \) and \( q \) be the propositions \( p \) : I bought a lottery ticket this week. \( q \) : I won the million dollar jackpot. Express each of these propositions as an English sentence. a) \( \neg p \) d) \( p \wedge q \) b) \( \mathrm{p} \vee \mathrm{q} \) e) \( p \leftrightarrow q \) c) \( \mathrm{p} \rightarrow \mathrm{q} \) f) \( p \rightarrow \neg q \) Chapter One: The Foundations (Logic and Proofs) Page 1 8) Determine whether each of these conditional statements is true or false. a) If \( 1+1=3 \), then unicorns exist. b) If \( 1+1=3 \), then dogs can fly. c) If \( 1+1=2 \), then dogs can fly. d) If \( 2+2=4 \), then \( 1+2=3 \). 9) Construct a truth table for each of these compound propositions. a) \( p \wedge \neg p \) b) \( p \vee \neg p \) c) \( (p \vee \neg q) \rightarrow q \) d) \( (p \vee q) \rightarrow(p \wedge q) \) e) \( (p \rightarrow q) \leftrightarrow(\neg q \rightarrow \neg p) \) f) \( (p \rightarrow q) \rightarrow(q \rightarrow p) \) 10) Construct a truth table for each of these compound propositions. a) \( \mathrm{p} \rightarrow \neg \mathrm{q} \) b) \( \neg \mathrm{p} \leftrightarrow \mathrm{q} \) c) \( (\mathrm{p} \rightarrow \mathrm{q}) \vee(\neg \mathrm{p} \rightarrow \mathrm{q}) \) d) \( (\mathrm{p} \rightarrow \mathrm{q}) \wedge(\neg \mathrm{p} \rightarrow \mathrm{q}) \) e) \( (p \leftrightarrow q) \vee(\neg p \leftrightarrow q) \) f) \( (\neg \mathrm{p} \leftrightarrow \neg q) \leftrightarrow(\mathrm{p} \leftrightarrow \mathrm{q} \) 11) Construct a truth table for each of these compound propositions. a) \( (p \vee q) \vee r \) b) \( (\mathrm{p} \vee \mathrm{q}) \wedge \mathrm{r} \) c) \( (p \wedge q) \vee r \) d) \( (p \wedge q) \wedge r \) e) \( (p \vee q) \wedge \neg r \) f) \( (p \wedge q) \vee \neg r \) 12) Construct a truth table for \( ((p \rightarrow q) \rightarrow r) \rightarrow s \). 13) Use truth tables to verify these equivalences. a) \( p \wedge T \equiv p \) b) \( p \vee \boldsymbol{F} \equiv p \) c) \( p \wedge \boldsymbol{F} \equiv \boldsymbol{F} \) d) \( p \vee T \equiv T \) e) \( p \vee p \equiv p \) f) \( p \wedge p \equiv p \) 14) Show that \( \neg(\neg p) \) and \( p \) are logically equivalent. 15) Use truth tables to verify the commutative laws a) \( p \vee q \equiv q \vee p \) b) \( p \wedge q \equiv q \wedge p \) 16) Use a truth table to verify the first De Morgan law \[ \neg(p \wedge q) \equiv \neg p \vee \neg q \] 17) Determine whether \( (\neg p \wedge(p \rightarrow q)) \rightarrow \neg q \) is a tautology. 18) Show that each of these conditional statements is a tautology by using truth tables. a) \( [\neg p \wedge(p \vee q)] \rightarrow q \) b) \( [(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r) \) c) \( [p \wedge(p \rightarrow q)] \rightarrow q \) d) \( [(p \vee q) \wedge(p \rightarrow r) \wedge(q \rightarrow r)] \rightarrow r \) 19) Show that \( \neg p \rightarrow(q \rightarrow r) \) and \( q \rightarrow(p \vee r) \) are logically equivalent. Chapter One: The Foundations (Logic and Proofs) Page 2 20) Use a direct proof to show that the sum of two even integers is even. 21) Use a direct proof to show that the product of two odd numbers is odd. 22) Prove that if \( n \) is a perfect square, then \( n+2 \) is not a perfect square. 23) Prove that if \( n \) is an integer and \( 3 n+2 \) is even, then \( n \) is even using a) a proof by contraposition.

          b) What time is it?
Safiirka Islaamka 10 Xaqiiqo Oo La Yaab Leh Oo Ku Saabsan Shukaansiga Sheikh Axmed...
(1) \( +\ldots-\ldots \)
e) The moon is made of green cheese.
7) Let \( p \) and \( q \) be the propositions
\( p \) : I bought a lottery ticket this week.
\( q \) : I won the million dollar jackpot.

Express each of these propositions as an English sentence.
a) \( \neg p \)
d) \( p \wedge q \)
b) \( \mathrm{p} \vee \mathrm{q} \)
e) \( p \leftrightarrow q \)
c) \( \mathrm{p} \rightarrow \mathrm{q} \)
f) \( p \rightarrow \neg q \)

Chapter One: The Foundations (Logic and Proofs)
Page 1
8) Determine whether each of these conditional statements is true or false.
a) If \( 1+1=3 \), then unicorns exist.
b) If \( 1+1=3 \), then dogs can fly.
c) If \( 1+1=2 \), then dogs can fly.
d) If \( 2+2=4 \), then \( 1+2=3 \).
9) Construct a truth table for each of these compound propositions.
a) \( p \wedge \neg p \)
b) \( p \vee \neg p \)
c) \( (p \vee \neg q) \rightarrow q \)
d) \( (p \vee q) \rightarrow(p \wedge q) \)
e) \( (p \rightarrow q) \leftrightarrow(\neg q \rightarrow \neg p) \)
f) \( (p \rightarrow q) \rightarrow(q \rightarrow p) \)
10) Construct a truth table for each of these compound propositions.
a) \( \mathrm{p} \rightarrow \neg \mathrm{q} \)
b) \( \neg \mathrm{p} \leftrightarrow \mathrm{q} \)
c) \( (\mathrm{p} \rightarrow \mathrm{q}) \vee(\neg \mathrm{p} \rightarrow \mathrm{q}) \)
d) \( (\mathrm{p} \rightarrow \mathrm{q}) \wedge(\neg \mathrm{p} \rightarrow \mathrm{q}) \)
e) \( (p \leftrightarrow q) \vee(\neg p \leftrightarrow q) \)
f) \( (\neg \mathrm{p} \leftrightarrow \neg q) \leftrightarrow(\mathrm{p} \leftrightarrow \mathrm{q} \)
11) Construct a truth table for each of these compound propositions.
a) \( (p \vee q) \vee r \)
b) \( (\mathrm{p} \vee \mathrm{q}) \wedge \mathrm{r} \)
c) \( (p \wedge q) \vee r \)
d) \( (p \wedge q) \wedge r \)
e) \( (p \vee q) \wedge \neg r \)
f) \( (p \wedge q) \vee \neg r \)
12) Construct a truth table for \( ((p \rightarrow q) \rightarrow r) \rightarrow s \).
13) Use truth tables to verify these equivalences.
a) \( p \wedge T \equiv p \)
b) \( p \vee \boldsymbol{F} \equiv p \)
c) \( p \wedge \boldsymbol{F} \equiv \boldsymbol{F} \)
d) \( p \vee T \equiv T \)
e) \( p \vee p \equiv p \)
f) \( p \wedge p \equiv p \)
14) Show that \( \neg(\neg p) \) and \( p \) are logically equivalent.
15) Use truth tables to verify the commutative laws
a) \( p \vee q \equiv q \vee p \)
b) \( p \wedge q \equiv q \wedge p \)
16) Use a truth table to verify the first De Morgan law
\[
\neg(p \wedge q) \equiv \neg p \vee \neg q
\]
17) Determine whether \( (\neg p \wedge(p \rightarrow q)) \rightarrow \neg q \) is a tautology.
18) Show that each of these conditional statements is a tautology by using truth tables.
a) \( [\neg p \wedge(p \vee q)] \rightarrow q \)
b) \( [(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r) \)
c) \( [p \wedge(p \rightarrow q)] \rightarrow q \)
d) \( [(p \vee q) \wedge(p \rightarrow r) \wedge(q \rightarrow r)] \rightarrow r \)
19) Show that \( \neg p \rightarrow(q \rightarrow r) \) and \( q \rightarrow(p \vee r) \) are logically equivalent.
Chapter One: The Foundations (Logic and Proofs)
Page 2
20) Use a direct proof to show that the sum of two even integers is even.
21) Use a direct proof to show that the product of two odd numbers is odd.
22) Prove that if \( n \) is a perfect square, then \( n+2 \) is not a perfect square.
23) Prove that if \( n \) is an integer and \( 3 n+2 \) is even, then \( n \) is even using
a) a proof by contraposition.

b) What time is it?
Safiirka Islaamka 10 Xaqiiqo Oo La Yaab Leh Oo Ku Saabsan Shukaansiga Sheikh Axmed...
(1) +…-…
e) The moon is made of green cheese.
7) Let p and q be the propositions
p : I bought a lottery ticket this week.
q : I won the million dollar jackpot.

Express each of these propositions as an English sentence.
a) p
d) p ∧ q
b) p∨q
e) p ↔ q
c) p→q
f) p → q

Chapter One: The Foundations (Logic and Proofs)
Page 1
8) Determine whether each of these conditional statements is true or false.
a) If 1+1=3, then unicorns exist.
b) If 1+1=3, then dogs can fly.
c) If 1+1=2, then dogs can fly.
d) If 2+2=4, then 1+2=3.
9) Construct a truth table for each of these compound propositions.
a) p ∧ p
b) p ∨ p
c) (p ∨ q) → q
d) (p ∨ q) →(p ∧ q)
e) (p → q) ↔( q → p)
f) (p → q) →(q → p)
10) Construct a truth table for each of these compound propositions.
a) p→q
b) p↔q
c) (p→q) ∨(p→q)
d) (p→q) ∧(p→q)
e) (p ↔ q) ∨( p ↔ q)
f) (p↔ q) ↔(p↔q
11) Construct a truth table for each of these compound propositions.
a) (p ∨ q) ∨ r
b) (p∨q) ∧r
c) (p ∧ q) ∨ r
d) (p ∧ q) ∧ r
e) (p ∨ q) ∧ r
f) (p ∧ q) ∨ r
12) Construct a truth table for ((p → q) → r) → s.
13) Use truth tables to verify these equivalences.
a) p ∧ T ≡ p
b) p ∨F≡ p
c) p ∧F≡F
d) p ∨ T ≡ T
e) p ∨ p ≡ p
f) p ∧ p ≡ p
14) Show that ( p) and p are logically equivalent.
15) Use truth tables to verify the commutative laws
a) p ∨ q ≡ q ∨ p
b) p ∧ q ≡ q ∧ p
16) Use a truth table to verify the first De Morgan law

(p ∧ q) ≡ p ∨ q

17) Determine whether ( p ∧(p → q)) → q is a tautology.
18) Show that each of these conditional statements is a tautology by using truth tables.
a) [ p ∧(p ∨ q)] → q
b) [(p → q) ∧(q → r)] →(p → r)
c) [p ∧(p → q)] → q
d) [(p ∨ q) ∧(p → r) ∧(q → r)] → r
19) Show that p →(q → r) and q →(p ∨ r) are logically equivalent.
Chapter One: The Foundations (Logic and Proofs)
Page 2
20) Use a direct proof to show that the sum of two even integers is even.
21) Use a direct proof to show that the product of two odd numbers is odd.
22) Prove that if n is a perfect square, then n+2 is not a perfect square.
23) Prove that if n is an integer and 3 n+2 is even, then n is even using
a) a proof by contraposition.

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