Answer to Question #176424 in Discrete Mathematics for Nicolas

Question

Let P (x): x²/2 =x

find the following and identify their truth values

1. (P) 1

2. (P) 2

3.∀n, P(n)

4.∃n, P(n)

Answer 1

1. $\displaystyle P(1): \frac{1^2}{2}= \frac{1}{2} \neq 1 \Rightarrow \text{False}$
2. $\displaystyle P(2): \frac{2^2}{2}= 2 =2 \Rightarrow \text{True}$
3. in verbs: for all n P(n). This is false, because we can give a contrexample (like P(1)).
4. in verbs: exists such a n that P(n). This is true, because as we saw for P(2) such n=2 indeed exists.

Answer: 1.F, 2. T, 3. F, 4. T.