**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

p ∨ q: p or q

p ∧ 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 ofthese 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 ∨ (¬p ∧ q)

Answer:

**Given:**

p: The election is decided

q: The votes have been counted.

**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.

**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.

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.