91影视

For a conditional statement to be true, if the hypothesis is true, then the conclusion must also be true.

The given statement is 鈥淚f $2\left(b+c\right)=2b+2c$, then $2+\left(b鈰�c\right)=\left(2+b\right)\left(2+c\right)$鈥�.

The hypothesis is the part of the statement following 鈥渋f鈥�, that is, 鈥�$2\left(b+c\right)=2b+2c$鈥� and the conclusion is the part of the statement following 鈥渢hen鈥�, that is, 鈥�$2+\left(b鈰�c\right)=\left(2+b\right)\left(2+c\right)$鈥�.

In general, for any two real numbers,$b$ and $c$,$2\left(b+c\right)=2b+2c$ holds due to the distributive property of multiplication over addition.

However, for any two real numbers,$b$ and $c$,$2+\left(b鈰�c\right)=\left(2+b\right)\left(2+c\right)$ does not hold as addition is not distributive over multiplication.

So, in general, the hypothesis of the given statement is true but the conclusion is false. This implies that the given statement is not always true.

Put$b=3$ and$c=5$ in$2\left(b+c\right)=2b+2c$ to get,

$\begin{array}{l}2\left(3+5\right)=2\left(3\right)+2\left(5\right)\\ 2\left(8\right)=6+10\\ 16=16\end{array}$

So, the hypothesis is true.

Put$b=3$ and$c=5$ in$2+\left(b鈰�c\right)=\left(2+b\right)\left(2+c\right)$ to get,

$\begin{array}{l}2+\left(3鈰�5\right)=\left(2+3\right)\left(2+5\right)\\ 2+15=\left(5\right)\left(7\right)\\ 17=35\end{array}$

This is a contradiction. Thus, the conclusion is false.

So,$b=3$ and$c=5$ is a counterexample of the given statement.