Essentially undergrad computing 101, however my lecturer doesnt make some aspects of the material clear.
I am asked to prove the following
Given that
$\rho_1 \subseteq A \times B$
$\rho_2 \subseteq B \times C$
$\rho_3 \subseteq C \times D$
prove that $dom(\rho_1\circ\rho_2) \subseteq dom(\rho_1)$
While I have the answers at my disposal and I understand the process, some of the symbolism and general thought process is unclear to me and inhibiting me from carrying out the arithmetic on my own.
i.e: By the definition of $dom$ we have that:
$dom(\rho_1\circ\rho_2) =\lbrace a|\exists b:(a,b)\in (\rho_1\circ\rho_2)\rbrace$
I've come to understand that that subset symbol actually means 'some or all' overlap as opposed to complete overlap which has cleared up much of my misunderstanding, but later in the question I'm asked something similar concerning the range of their cartesian product, but instead of
$\lbrace a|...\rbrace$
I instead get
$\lbrace b|...\rbrace$
When I thought that the vertical line meant 'such that' just like the ':'. Or does that region of the bracket correspond to something analogous to inputs and outputs of the set?
You haven't stated a definition of $\rho_1 \circ \rho_2$. A guess would be that it is $$ \rho_1 \circ \rho_2 = \{(a,c) \in A \times C \mid \exists b \in B, (a,b) \in \rho_1 \wedge (b,c) \in \rho_2 \}. $$ (Many people would denote this $\rho_2 \circ \rho_1$, but since you wrote $\rho_1 \circ \rho_2$, I will do so as well.)
Then the domain of this relation is $$ \begin{split} \operatorname{dom}(\rho_1 \circ \rho_2) &= \{ a \in A \mid \exists c \in C, (a,c) \in \rho_1 \circ \rho_2 \} \\ &= \{ a \in A \mid \exists c \in C, \exists b \in B, (a,b) \in \rho_1 \wedge (b,c) \in \rho_2 \} \\ &= \{ a \in A \mid (\exists b \in B)( (a,b) \in \rho_1 \wedge \exists c \in C, (b,c) \in \rho_2) \} \\ & \subseteq \{a \in A \mid (\exists b \in B) ((a,b) \in \rho_1) \} \\ & = \operatorname{dom}(\rho_1) . \end{split} $$