1. ## Relations

Prove:

$\displaystyle R\circ (S\cap T)\subseteq R \circ S \cap R \circ T$

and an example where inclusion is strict

$\displaystyle R \circ S\cap T \subseteq R\circ (S\cap R^-^1 \circ T)$

and an example where inclusion is strict

I'm not understanding the composition of the intersection

2. Originally Posted by gutnedawg
Prove:
$\displaystyle R\circ (S\cap T)\subseteq R \circ S \cap R \circ T$

and an example where inclusion is strict
$\displaystyle R \circ S\cap T \subseteq R\circ (S\cap R^-^1 \circ T)$
and an example where inclusion is strict
$\displaystyle R\circ (S\cap T)\subseteq (R \circ S) \cap (R \circ T)$
Written that way the statement is true.
I have no idea what $\displaystyle R \circ S\cap T \subseteq R\circ (S\cap R^-^1 \circ T)$ could mean.
Have you miss-written that?

3. I know it is true I need to prove it and no I did not miss write the last part

The second one you wrote is meaningless.

5. I have prove one of the modular laws for realtions R, S, T

$\displaystyle R \circ S \cap T \subseteq R \circ (S\cap R^-^1 \circ T)$

or

$\displaystyle R \cap S \circ T \subseteq (R\circ T^-^1 \cap S) \circ T$

6. Originally Posted by gutnedawg
I have prove one of the modular laws for realtions R, S, T
$\displaystyle R \circ S \cap T \subseteq R \circ (S\cap R^-^1 \circ T)$
or $\displaystyle R \cap S \circ T \subseteq (R\circ T^-^1 \cap S) \circ T$
The way both of those are written makes totally meaningless.
Look at the way I used parentheses in post #2.
Without parentheses there is no way to know what goes with what.

In the second one it makes no sense to have $\displaystyle R^{-1}$ in it.
If you cannot provide a readable question, the we cannot help.

Now here is the proof of a standard question.
If $\displaystyle (a,b) \in R \circ \left( {S \cap T} \right)$ then $\displaystyle \left( {\exists c} \right)\left[ {(a,c) \in \left( {S \cap T} \right)\;\& \;(c,b) \in R} \right]$.

That also means $\displaystyle \left[ {\left( {(a,c) \in S\;\& \;(c,b) \in R} \right)\;\& \;\left( {(a,c) \in T\;\& \;(c,b) \in R} \right)} \right]$.

Or $\displaystyle (a,b) \in \left( {R \circ S} \right) \cap \left( {R \circ T} \right)$.

7. $\displaystyle R^-^1$

is the inverse relation
$\displaystyle \{(b,a) : (a,b) \epsilon R\}$