Exercises 1.1 3. What is the negation of each of these proposition ? a) Today is Friday. Ans: “ Today is not Thuesday ”.
b) There is no pollution in new Jersey. Ans: “ There is is pollution in new Jersey” . c) 2+1=3. Ans: “ 2+1 ≠ 3 ”. d) The summer in Maine is hot and sunny. Ans: “ The summer in Maine is not hot and sunny ”.
5. Let p and q be the proposition “Swimming is the New Jersey shore is allowed” and “Sharks have been spotted near the shore ,” respectively . Express of these compound proposition as an English sentence.
a) ¬ q Ans: Sharks have not been spotted near the shore . b) p ∧q Ans: Swimming is the New Jersey shore is allowed and sharks have been spotted near the shore . c) ¬ p ∨q Ans: Swimming is the New Jersey shore is not allowed or sharks have been spotted near the shore . d) p → ¬ q Ans: If swimming swimming is the New Jersey Jersey shore is allowed, then then sharks have not been spotted near the shore . e) ¬ q → p Ans: If sharks have not been spotted near the shore, then swimming is the New Jersey shore is allowed . f) ¬ p → ¬ q Ans: If Swimming is the New Jersey shore is not allowed then sharks ha ve not been spotted near the shore . g) p ↔ ¬ q Ans: Swimming is the New Jersey shore is allowed if and only if sharks have not been spotted near the shore
h) ¬ p ∧(p ∨ ¬ q) Ans: Swimming is the New Jersey shore is not allowed,and either swimming is the N ew Jersey shore is allowed or sharks have not b een spotted near the shore .
7. Let p and q be the propositions p : It is below freezing. q : It is snowing. Write these propositions using p and q and logical connectives.
a) It is below freezing and snowing. Ans : p ∧q b) It is below freezing but not snowing. Ans : p ∧¬ q c) It is not below freezing and it is not snowing. Ans : ¬ p ∧¬ q d) It is either snowing or below freezing (or both). Ans : p ∨q e) If it t is below freezing, it is also snowing. Ans : p → q f) It is either below freezing or it is snowing, but it is not snowing if it is below freezing. Ans : ( p ∨q ) ∧ ( p → ¬ q ) g) That it is below freezing is necessary and sufficient for it to be snowing. Ans : q ↔ p
9. Let p and q be the propositions p : You drive over 65 miles per hour. q : You get a speeding ticket. Write these propositions using p and q and logical connectives.
a) You do not drive over 65 miles per hour. Ans : ¬ p b) You drive over 65 miles per hour , but you do not get a speeding ticket. Ans : p ∧¬ q c) You will get a speeding ticket if you drive over 65 miles per hour.
Ans : p → q d) If you do not drive over 65 miles per hour , then you will not get a speeding ticket. Ans : ¬ p → ¬ q e) Driving over 65 miles per hour is sufficient for getting a speeding ticket. Ans : p → q f) You get a speeding ticket , but you do not drive over 65 miles per hour. Ans : q ∧¬ p
g) Whenever you get a speeding ticket , you are driving over 65 miles per hour. Ans : q → p 34. Evaluate each of these expressions. a) 11000 ∧( 01011 ∨11011 ) b) ( 01111 ∧10101 ) ∨01000 c) ( 01010
