# Thread: Proof

1. ## Proof

Prove the Following:

R ◦ (S ∩ T) ⊆ (R ◦ S) ∩ (R ◦ T)

I think this is trying to show that this proof is trying to show that the compositions of relations are associative.

2. You know that $~(a,b) \in R \circ (S \cap T)\; \Rightarrow \;\left( {\exists c} \right)\left[ {(a,c) \in (S \cap T) \wedge (c,b) \in R} \right]$.

Can you finish?

3. Can we say that since the pair (c,b) is an element of R and the pair (a,c) is an element of (S ∩ T) then the pair (a,c) is an element of both?

4. Originally Posted by rnuravvk
Can we say that since the pair (c,b) is an element of R and the pair (a,c) is an element of (S ∩ T) then the pair (a,c) is an element of both?
So $(a,c) \in S \wedge (c,b) \in R\;\& \,(a,c) \in T \wedge (c,b) \in R$. What does that imply?

5. That the composition of S with T and T with R are intersecting one another?

6. Originally Posted by rnuravvk
That the composition of S with T and T with R are intersecting one another?
I must ask you: "Do you understand any of this"?
Are you just making wild guesses?
If not, please give some reasons.

7. So I re-thought it over and think this is it. The pair (a,b) is an element of S,T and R

8. Originally Posted by rnuravvk
So I re-thought it over and think this is it. The pair (a,b) is an element of S,T and R
You are guessing are you not?
Where are the compositions?
Do you understand what any of this means?