MAT-270 Written Assn 1
Written Assignment 1
Section 1.1:
8b). I bought a lottery ticket this week OR I won the million dollar jackpot on Friday.
e). I bought a lottery ticket this week IF AND ONLY IF I won the million dollar jackpot on Friday.
f).If I did not buy a lottery ticket this week,then I did not win the million dollar lottery on Friday.
12f).You have the flu AND you miss the final examination,OR if you do not miss the final examination AND ytou pass the course.
18a). The conditional statement:if 1+1=3 then unicorns exist is true.
b). The conditional statement:if 1+1=3,then dogs can fly is true.
c). The conditional statement:if 1+1=2,then dogs can fly is false.
d). The conditional statement:if 2+2=4,then 1+2=3 is true.
32a).
e). 34e). 36f). Section 1.3:
10c).
So,[p∧(p→q)]→qis a tautology.
12b). [(p → q) ∧ (q → r)] → (p → r)
≡ [(¬p ∨ q) ∧ (¬q ∨ r)] → (¬p ∨ r) (Rule of Implication)
≡ ¬[(¬p ∨ q) ∧ (¬q ∨ r)] ∨ (¬p ∨ r) (Rule of Implication)
≡ [¬(¬p ∨ q) ∨ ¬(¬q ∨ r)] ∨ (¬p ∨ r) (DeMorgan’s Law)
≡ [(¬(¬p) ∧ ¬q) ∨ (¬(¬q) ∧ ¬r)] ∨ (¬p ∨ r) (DeMorgan’s Law)
≡ [(p ∧ ¬q) ∨ (q ∧ ¬r)] ∨ (¬p ∨ r) (Law of Double Negation)
≡ (p ∧ ¬q) ∨ (q ∧ ¬r) ∨ (¬p) ∨ r (Associative Law)
≡ (¬p) ∨ (p ∧ ¬q) ∨ r ∨ (q ∧ ¬r) (Commutative Law)
≡ (¬p ∨ (p ∧ ¬q)) ∨ (r ∨ (q ∧ ¬r)) (Associative Law)
≡ [(¬p ∨ p) ∧ (¬p ∨ ¬q)] ∨ [(r ∨ q) ∧ (r ∨ ¬r)] (Distributive Law)
≡ [T ∧ (¬p ∨ ¬q)] ∨ [(r ∨ q) ∧ T] (Negative Law)
≡ (¬p ∨ ¬q) ∨ (r ∨ q) (Identity Law)
≡ (¬p) ∨ (¬q) ∨ r ∨ q (Assocaitive Law)
≡ (¬p) ∨ r ∨ (¬q) ∨ q (Commutative Law)
≡ (¬p ∨ r) ∨ (¬q ∨ q) (Associative Law)
≡ (¬p ∨ r) ∨ T (Negation Law)
≡ T (Domination Law) 16). So,p↔q and (p∧q)∨(¬p∧¬q)are equivalent.
24).
So,(p→q)∨(p→r) and p→(q∨r) are logically equivalent.
28).
So,p↔q and ¬p↔¬q are logically equivalent. Section 1.4:
8b). Every rabbit hops.
d). There is an animal rabbit that hops.
12d).∃xQ(x)is true because x+1>2x ∃ some x
f).∃x¬Q(x)is true because x+1>2x is true for
Section 1.1:
8b). I bought a lottery ticket this week OR I won the million dollar jackpot on Friday.
e). I bought a lottery ticket this week IF AND ONLY IF I won the million dollar jackpot on Friday.
f).If I did not buy a lottery ticket this week,then I did not win the million dollar lottery on Friday.
12f).You have the flu AND you miss the final examination,OR if you do not miss the final examination AND ytou pass the course.
18a). The conditional statement:if 1+1=3 then unicorns exist is true.
b). The conditional statement:if 1+1=3,then dogs can fly is true.
c). The conditional statement:if 1+1=2,then dogs can fly is false.
d). The conditional statement:if 2+2=4,then 1+2=3 is true.
32a).
e). 34e). 36f). Section 1.3:
10c).
So,[p∧(p→q)]→qis a tautology.
12b). [(p → q) ∧ (q → r)] → (p → r)
≡ [(¬p ∨ q) ∧ (¬q ∨ r)] → (¬p ∨ r) (Rule of Implication)
≡ ¬[(¬p ∨ q) ∧ (¬q ∨ r)] ∨ (¬p ∨ r) (Rule of Implication)
≡ [¬(¬p ∨ q) ∨ ¬(¬q ∨ r)] ∨ (¬p ∨ r) (DeMorgan’s Law)
≡ [(¬(¬p) ∧ ¬q) ∨ (¬(¬q) ∧ ¬r)] ∨ (¬p ∨ r) (DeMorgan’s Law)
≡ [(p ∧ ¬q) ∨ (q ∧ ¬r)] ∨ (¬p ∨ r) (Law of Double Negation)
≡ (p ∧ ¬q) ∨ (q ∧ ¬r) ∨ (¬p) ∨ r (Associative Law)
≡ (¬p) ∨ (p ∧ ¬q) ∨ r ∨ (q ∧ ¬r) (Commutative Law)
≡ (¬p ∨ (p ∧ ¬q)) ∨ (r ∨ (q ∧ ¬r)) (Associative Law)
≡ [(¬p ∨ p) ∧ (¬p ∨ ¬q)] ∨ [(r ∨ q) ∧ (r ∨ ¬r)] (Distributive Law)
≡ [T ∧ (¬p ∨ ¬q)] ∨ [(r ∨ q) ∧ T] (Negative Law)
≡ (¬p ∨ ¬q) ∨ (r ∨ q) (Identity Law)
≡ (¬p) ∨ (¬q) ∨ r ∨ q (Assocaitive Law)
≡ (¬p) ∨ r ∨ (¬q) ∨ q (Commutative Law)
≡ (¬p ∨ r) ∨ (¬q ∨ q) (Associative Law)
≡ (¬p ∨ r) ∨ T (Negation Law)
≡ T (Domination Law) 16). So,p↔q and (p∧q)∨(¬p∧¬q)are equivalent.
24).
So,(p→q)∨(p→r) and p→(q∨r) are logically equivalent.
28).
So,p↔q and ¬p↔¬q are logically equivalent. Section 1.4:
8b). Every rabbit hops.
d). There is an animal rabbit that hops.
12d).∃xQ(x)is true because x+1>2x ∃ some x
f).∃x¬Q(x)is true because x+1>2x is true for