You & maths

The Foundations: Logic and Proofs – Exercises from 11 to 15

11. Let p and q be the propositions “Swimming at the New Jersey shore is allowed” and “Sharks have been spotted near the shore,” respectively. Express each of these compound propositions as an English sentence.

a) ¬q                          b) p ∧ q                     c) ¬p q

d) p → ¬q                  e) ¬q → p                  f) ¬p → ¬q

g) p ↔ ¬q                h) ¬p (p ¬q)

Answer:

Given:

p: Swimming at the New Jersey shore is allowed.

q: Sharks have been spotted near the shore.

INTERPRETATION SYMBOLS

 q: p or q

 q: p and q

p → q: if p then q

p ↔ q: p if only if q

SOLUTION

Determine the English sentences by replacing p and q by their given in the above interpretations (and adjust the sentence to form a proper English sentence as needed):

(a) Sharks have not been spotted near the shore.

(b) Swimming at the New Jersey shore is allowed and sharks have been spotted near the shore.

(c) Swimming at the New Jersey is not allowed or sharks have been spotted near the shore.

(d) If swimming at the New Jersey shore is allowed, then sharks have not been spotted near shore.

(e) If sharks have not been spotted near shore, then swimming at New Jersey shore is allowed.

(f) If swimming at the New Jersey shore is not allowed, then sharks have not been spotted near the shore.

(g) Swimming at the New Jersey shore is allowed if and only if sharks have not been spotted near the shore.

(h) Swimming at the New Jersey shore is not allowed and, swimming at the New Jersey shore is either allowed or sharks have not been spotted near the shore.

Note: The word “either” replaced the parentheses in the given proposition.

12. Let p and q be the propositions “The election is decided” and “The votes have been counted,” respectively. Express each of
these compound propositions as an English sentence.

a) ¬p                           b) p q

c) ¬p ∧ q                    d) q p

e) ¬q → ¬p                f) ¬p → ¬q

g) p ↔ q                   h) ¬q (¬∧ q)

Answer:

Given:

p: The election is decided

q: The votes have been counted.

Reference INTERPRETATION SYMBOLS 11

SOLUTION

Determine the sentences by replacing p and q by their given sentences in the above (and the sentence to
form a proper English sentence as needed) :

(a) The election is not decided.

(b) election decided or the votes have been been counted.

(c) The election is not decided and the votes have been counted.

(d) If votes have been counted, then the election is decided.

(e) If the votes have not been counted, then the election not decided.

(f) If the election is not decided, then votes have not counted.

(g) The election is decided if and only if the votes have been counted.

(h) The votes have not been counted or, the election is not decided and the votes have been counted.

Note: We replace the brackets by a comma “,” between the statement (to represent the brackets).

13. 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 (including negations).

a) It is below freezing and snowing.

b) It is below freezing but not snowing.

c) It is not below freezing and it is not snowing.

d) It is either snowing or below freezing (or both).

e) If it is below freezing, it is also snowing.

f) Either it is below freezing or it is snowing, but it is not snowing if it is below freezing.

g) That it is below freezing is necessary and sufficient for it to be snowing.

Answer:

Given:

p: It is below freezing

q: It is snowing.

Reference INTERPRETATION SYMBOLS 11

SOLUTION

Replace the propositions of p and q by their symbol in the given propositions,then use interpretation of the symbols.

(a) The given proposition states: p and q, or equivalently p  q.

(b) The given proposition states: p and not q, or equivalently p  —q.

(c) The given proposition states: not p and not q, or equivalently -p  -q

(d) The given proposition states: p or q, or equivalently p  q.

(e) The given proposition states: if p, then, or equivalertly p→ q.

(f) The given proposition states: p or q, and if p then not q, or equivalently (p  q)  (p → -q)

Note: The comma “,” divides the sentence in two parts and there should be brackets in the
proposition about each part of the sentence.

(g) “Necessary and sufficient” means the Same as “if and only if”. The given proinsition then
states: p if and only if q, or equivalertly p ↔ q.

14. Let p, q, and r be the propositions

p: You have the flu.

q: You miss the final examination.

r: You pass the course.

Express each of these propositions as an English sentence.

a) p → q                                            b) ¬q ↔ r

c) q → ¬r                                           d) p ∨ q ∨ r

e) (p → ¬r) ∨ (q ¬r)                   f) (p ∧ q) ∨ (¬q r)

Answer:

a) If you have flu then you will miss the final examination.

b) You won’t miss the final examination if and only if you pass the course.

c) If you miss the examination then you will be failing the course

d) You have the flu OR you miss the final examination OR you pass the course.

e) If you have the flu then you’ll not pass the course OR If you miss the final examination then you’ll fail the course

f) You have the flu and you miss the examination OR You will not miss the final examination and you pass the course

15. 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 (including negations).

a) You do not drive over 65 miles per hour.

b) You drive over 65 miles per hour, but you do not get a speeding ticket.

c) You will get a speeding ticket if you drive over 65 miles per hour.

d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.

e) Driving over 65miles per hour is sufficient for getting a speeding ticket.

f) You get a speeding ticket, but you do not drive over 65 miles per hour.

g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.

Answer:

Given:

 p: You drive over 65 miles per hour.

q: You get a speeding ticket.

Reference INTERPRETATION SYMBOLS 11

SOLUTION

Replace the propositions of p and q by their symbol in the given propositions, then use the interpretation of the
symbols.

(a) The given proposition states: not p, or equivalently —p.

(b) Tho given proposition states: p and not q, or equivalently p  —q.

(c) The given proposition states: if p then q, or equivalently p q.

(d) The given proposition states: if not p then not q, or equivalently —p → —q

(e) “Sufficient” means the same as the if-then statement. The given proposition states: if p, then q, or equivalently p → q.

(f) The given proposition states: q and not p, or equivalently q —p.

(g) We could rewrite the sentence as: If you get a speeding ticket, then are driving are 63 miles per hour. The given proposition then states: if q then p, or quivalently q  p.