Question

Let\({\rm{p}}\left( {\rm{x}} \right)\) denote the statement “ \({\rm{x}} \le 4\)” what are these truth values

(a) p(0)

(b) p(4)

(c) p(6)

Short answer

The truth values of $\mathrm{p}\left(0\right),\mathrm{p}\left(4\right)$ and $\mathrm{p}\left(6\right)$ are true, true, and false respectively. For this, put the value of $\mathbf{x}$ in the given condition such as $\mathbf{x}\mathbf{\le }\mathbf{4}$and check for the result.

Step-by-step solution

Definition of the truth value

A truth valuein a truth table indicates the truth ($\mathbf{T}$or$\mathbf{1}$) or falsity ($\mathbf{F}$or $\mathbf{0}$) of a specified proposition or statement.

To find the truth values of given propositions.

(a) $\mathrm{p}\left(0\right)$

The given condition is $\mathrm{p}\left(\mathrm{x}\right)=\mathrm{x}\le 4$

The given proposition is $\mathrm{p}\left(0\right)$.

That means, $0\le 4$

Therefore, the statement $\mathrm{p}\left(0\right)$is true as 0is less than4.

(b) $\mathrm{p}\left(4\right)$

The given condition is $\mathrm{p}\left(\mathrm{x}\right)=\mathrm{x}\le 4$

The given proposition is $\mathrm{p}\left(4\right)$.

That means, $4\le 4$

Therefore, the statement $\mathrm{p}\left(4\right)$is true as $4$is equal to 4.

(c) $\mathrm{p}\left(6\right)$

The given condition is$\mathrm{p}\left(\mathrm{x}\right)=\mathrm{x}\le 4$

The given proposition is $\mathrm{p}\left(6\right)$.

That means,$\mathrm{p}\le 4$

Therefore, the statement $\mathrm{p}\left(6\right)$is false as $6$is not less than $4$.

Therefore, the truth values of $\mathrm{p}\left(0\right),\mathrm{p}\left(4\right)$and $\mathrm{p}\left(6\right)$are true, true, and false respectively.