# 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
Science Advisor
Homework Helper
Education Advisor
Is there any $x \in S$ such that P(x) is false?

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving