Question

P(x): x²/2 = x

Find the following then identify their truth values.

1. P (1)
2. P (2)
3. An, P(n)
4. En, P(n)

Answer 1

First of all, solve the equation: $\frac{x^2}{2}=x$. The latter can be written as: $(\frac12x-1)x=0$. Solutions of the latter are $x=2$ and $x=0.$ Thus, we receive the following:

1. $P(1)$ is false. 2. $P(2)$ is true. 3. $\forall n\,\, P(n)$ is false (not all numbers\natural numbers are solutions). 4. $\exists n\,\,P(n)$ is true, since $x=2$ is a solution.