Question

Let p be the proposition “I will do every exercise in this book” and q be the proposition “I will get an “A” in this course.” Express each of these as a combination of p and q.

a) I will get an “A” in this course only if I do every exercise in this book.

b) I will get an “A” in this course and I will do every exercise in this book.

c) Either I will not get an “A” in this course or I will not do every exercise in this book.

d) For me to get an “A” in this course it is necessary and sufficient that I do every exercise in this book.

Short answer

(a)\(q \to p\)

(b) \(q \wedge p\)

(c) \(\neg q \vee \neg p\)

(d) \(q \leftrightarrow p\)

Step-by-step solution

Introduction

A proposition is a statement that expresses an idea, a recommendation, or a plan.

(a) part presentation

Here the propositions are:

\(p\): I will do every exercise in this book.

\(q\): I will get an A in this course.

It can be seen that the statement “I will get an A in this course” is possible only if the statement “I will do every exercise in this book” is valid.

Thus, the second statement is conditional to make the first true.

Expressing mathematically the condition will be:

\(q \to p\)

(b) part presentation

I will get an A in this course and I will do every exercise in this book.

Here the propositions are:

\(p\): I will do every exercise in this book.

\(q\): I will get an A in this course.

It can be seen that the statement “I will get an A in this course” and the statement “I will do every exercise in this book” are combined by an operator.

Expressing mathematically the condition will be:

\(q \wedge p\)

(c) part presentation

Either I will not get an A in this course or I will not do every exercise in this book.

Here the propositions are:

\(p\): I will do every exercise in this book.

\(q\): I will get an A in this course.

It can be seen that the statement “I will not get an A in this course” and the statement “I will not do every exercise in this book” are both negation statements and both are connected with OR operator.

Expressing mathematically the condition will be:

\(\neg q \vee \neg p\)

(d) part presentation

For me to get an A in this course it is necessary and sufficient that I do every exercise in this book.

Here the propositions are:

\(p\): I will do every exercise in this book.

\(q\): I will get an A in this course.

It can be seen that the statement “I will not get an A in this course” and the statement “I will not do every exercise in this book” are bi-conditional here. In alternate words, it implies the condition if and only if.

Mathematically,

\(q \leftrightarrow p\)

Thus, the required result is found.