Chapter 7, Problem 1RE

To determine

To express: The proposition $\sim q$ as an English sentence and determine whether it is true or false.

Answer to Problem 1RE

The proposition $\sim q$ can be written as $\underset{\_}{{\text{3}}^{\text{4}}{\text{ is not less than 4}}^{\text{3}}}$ and it is $\underset{\_}{\text{true}}$.

Explanation of Solution

Given:

The proposition are p: “${\text{2}}^3<3^2$” q: “${\text{3}}^4<4^3$”.

Definition used:

Negation:

If p is a proposition, then the proposition $\sim p$, not p, the negation of p, is false if p is true and true if p is false.

Description:

Given that, the proposition is q: ${\text{3}}^4<4^3$.

It can be translated to a sentence as, “${\text{3}}^4$ is less than $4^3$.

The negation of q ($\sim q$) can be written as “${\text{3}}^{\text{4}}\ge {\text{4}}^{\text{3}}$”.

Now check whether the proposition “${\text{3}}^{\text{4}}\ge {\text{4}}^{\text{3}}$” is true or false as follows.

$\begin{array}{l}{\text{3}}^4\ge 4^3\\ 81\ge 64\end{array}$

It is known that 81 is greater than 64. Thus, q is false and $\sim q$ is true.

Thus, the proposition $\sim q$ can be written as $\underset{\_}{{\text{3}}^{\text{4}}{\text{ is not less than 4}}^{\text{3}}}$ and it is $\underset{\_}{\text{true}}$.