# Thread: Math Logic: Let A, B, C and D be sets, and R ⊆ A x B, S ⊆ B x C, T ⊆ C x D. Show that

1. ## Math Logic: Let A, B, C and D be sets, and R ⊆ A x B, S ⊆ B x C, T ⊆ C x D. Show that

Suppose that A, B, C and D are sets, R is a relation between A and B
(i.e., R ⊆ A x B), S is a relation between B and C and T is a relation
between C and D. Show the following:
(a) R ¤ (S ¤ T) = (R ¤ S) ¤ T.
(b) (R ¤ S)⁻¹ = S⁻¹ ¤ R⁻¹ .

2. ## Re: Math Logic: Let A, B, C and D be sets, and R ⊆ A x B, S ⊆ B x C, T ⊆ C x D. Show

Originally Posted by habsfan31
Suppose that A, B, C and D are sets, R is a relation between A and B
(i.e., R ⊆ A x B), S is a relation between B and C and T is a relation
between C and D. Show the following:
(a) R ¤ (S ¤ T) = (R ¤ S) ¤ T.
(b) (R ¤ S)⁻¹ = S⁻¹ ¤ R⁻¹ .
Cannot help if you don't tell about notation.
Does "R ¤ S" mean composition"

3. ## Re: Math Logic: Let A, B, C and D be sets, and R ⊆ A x B, S ⊆ B x C, T ⊆ C x D. Show

Yup, sorry bout that

4. ## Re: Math Logic: Let A, B, C and D be sets, and R ⊆ A x B, S ⊆ B x C, T ⊆ C x D. Show

Originally Posted by habsfan31
Yup, sorry bout that
If think then you have it backwards.
$\displaystyle S\circ R$ exists but $\displaystyle R\circ S$ does not.
Because $\displaystyle R$ relates $\displaystyle A\to B$ and $\displaystyle S$ relates $\displaystyle B\to C$.
So it must be $\displaystyle S\circ R$.
Unless your text material is completely non-standard.
It is usual for compositions to read right to left.

Post Script
Do you understand what I have posted?
If $\displaystyle {\mathcal{G}}\subseteq (U\times V)~\&~ {\mathcal{H}}\subseteq (V\times W)$ then if $\displaystyle U\ne W$ then $\displaystyle \mathcal{H}\circ \mathcal{G}$
exists BUT $\displaystyle \mathcal{G}\circ \mathcal{H}$ may not.

By definition $\displaystyle (u,w)\in \mathcal{H}\circ \mathcal{G} \iff \left( \exists v \right)[(u,v)\in \mathcal{G}~\&~(v,w)\in \mathcal{H}]$.

That means that $\displaystyle \text{Dom}(\mathcal{H})\subseteq\text{Img}(\mathca l{G}).$