# Set Theory Logic Question

#### knowLittle

1. The problem statement, all variables and given/known data
In each of the two following open sentences P(x) and Q(x) over a domain S are given.
Determine all $x \in S$ for which P(x) → Q(x) is a true statement.

$P(x): x \in [-1, 2]; Q(x): x^{2} \leq 2; S=[-1,1]$

2. Relevant equations
According to truth values for →:
a b a-> b
0 0 1
0 1 1
1 0 0
1 1 1
3. The attempt at a solution
If I can prove that P is False, then I will always get a T value for ->
Can I just say $x \in [-1,1]$, this would literally mean that statement P is false.
Could this count?

#### knowLittle

Would this mean that the statement is true vacuously?

#### vela

Staff Emeritus
Homework Helper
Is there any $x \in S$ such that P(x) is false?

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