Set Theory and Predicate Calculus?

[thename1000]

Set Theory and Predicate Calculus (12 points)
Given: P ⊆ Q
Q ⊆ (S ∩ T)
S ⊆ (R ∪ T^c)
x(sub)1 ∈ P
Use predicate calculus to prove x(sub)1 ∈ R.

Studying for a test but I don't have this worked out for me. I honestly don't even know where to start. I know what union, intersect, etc and all the symbols mean I'm just bad at the Predicate Calculus.

 

[g_edgar]

I think you need help from someone who knows that particular textbook.

 

[thename1000]

I think you need help from someone who knows that particular textbook.

Oh really its that specific? :( too bad

 

[honestrosewater]

Nah, I might be able to help. I will look at it after class.

Why not start by drawing a picture (e.g., a Venn diagram) to see what a model of these sentences must look like? I find that pictures are especially helpful at suggesting proofs by contradiction.

 

[HallsofIvy]

Set Theory and Predicate Calculus (12 points)
Given: P ⊆ Q
Q ⊆ (S ∩ T)
S ⊆ (R ∪ T^c)
x(sub)1 ∈ P
Use predicate calculus to prove x(sub)1 ∈ R.

Studying for a test but I don't have this worked out for me. I honestly don't even know where to start. I know what union, intersect, etc and all the symbols mean I'm just bad at the Predicate Calculus.

If $x_1\in P$ then, by the first line, $x_1\in Q$. By the second line $x_1\in S$ and in T. By the third line then, $x_1\in R$ or $x_1\in T^c$. But since $x_1\in T$, it can't be in $T^c$. Therefore $x_1\in R$.

Now all you have to do is express that in predicate calculus!