Question

Find the negations of these statements.

a) If it snows today, then I will go skiing tomorrow.

b) Every person in this class understands mathematical induction.

c) Some students in this class do not like discrete mathematics.

d) In every mathematics class there is some student who falls asleep during lectures.

Short answer

a) \(P\left( x \right)\): If it snows today, then I will go skiing tomorrow.

\(\neg P\left( x \right)\): If it will snow today, I will not go skiing tomorrow.

b) \(P\left( x \right)\): Every person in this class understands mathematical induction.

\(\neg P\left( x \right)\): Every person in this class does not understand mathematical induction.

c) \(P\left( x \right)\): Some students in this class do not like discrete mathematics.

\(\neg P\left( x \right)\): All students in this class like discrete mathematics.

d) \(P\left( x \right)\): In mathematics class, there is some student who falls asleep during lectures.

\(\neg P\left( x \right)\): In mathematics class in which all the students stay awake during lectures.

Step-by-step solution

Introduction

The negation operator is a unary operator that alters the truth value of a proposition p when applied to it.

Negations

Write the negations of the following statements as:

a)

\(P\left( x \right)\): If it snows today, then I will go skiing tomorrow.

\(\neg P\left( x \right)\): If it will snow today, I will not go skiing tomorrow.

b)

\(P\left( x \right)\): Every person in this class understands mathematical induction.

\(\neg P\left( x \right)\): Every person in this class does not understand mathematical induction.

c)

\(P\left( x \right)\): Some students in this class do not like discrete mathematics.

\(\neg P\left( x \right)\): All students in this class like discrete mathematics.

d)

\(P\left( x \right)\): In mathematics class, there is some student who falls asleep during lectures.

\(\neg P\left( x \right)\): In mathematics class in which all the students stay awake during lectures.