MHB 411.1.3.15 Prove A\cap(B/C)=(A\cap B)/(A\cap C)

[karush]

$\tiny{411.1.3.15}$

$\text{15. Prove $A\cap(B/C)=(A\cap B)/(A\cap C)$}$
and
$\textsf{19. Let $f:A \to B$ and $g:B \to C$ be in invertable mappings;} \\
\text{that is, mappings such that $f^{-1}$ and $g^{-1}$ exist}\\
\text{Show that $\textit{$(g \, o \, f)^{-1}$}$}$

ok I am starting to do this and want to take a class in it starting 082018
so hope mhb can help me get a head start

text pdf is on pg15 #15 and #19
http://text:http://abstract.ups.edu/download/aata-20150812.pdf

 

[Chris L T521]

$\tiny{411.1.3.15}$

$\text{15. Prove $A\cap(B/C)=(A\cap B)/(A\cap C)$}$
and
$\textsf{19. Let $f:A \to B$ and $g:B \to C$ be in invertable mappings;} \\
\text{that is, mappings such that $f^{-1}$ and $g^{-1}$ exist}\\
\text{Show that $\textit{$(g \, o \, f)^{-1}$}$}$

ok I am starting to do this and want to take a class in it starting 082018
so hope mhb can help me get a head start

text pdf is on pg15 #15 and #19
http://text:http://abstract.ups.edu/download/aata-20150812.pdf

Hi karush,

For 15., consider the following definitions:

\(A\cap B = \{x\mid x\in A\wedge x\in B\}\)

\(A\setminus B = \{x\mid x\in A \wedge x\notin B\}\)

It follows that

\(\begin{aligned}A\cap(B\setminus C) &= \{x\mid x\in A\wedge(x\in B \wedge x\notin C)\}\\ &= \{x\mid (x\in A\wedge x\in B)\wedge (x\in A\wedge x\notin C)\}\\ &=\{x\mid (x\in A\cap B)\wedge (x\notin A\cap C)\}\\ &= (A\cap B)\setminus (A\cap C)\end{aligned}\)

You might need to be more formal with your proof, but this should give you enough of an idea as to how to construct a formal proof of this set identity.

For 19., we note that \(f:A\rightarrow B\) and \(g:B\rightarrow C\); Hence \(g\circ f:A\rightarrow C\). To show that \((g\circ f)^{-1}= f^{-1}\circ g^{-1}\), you want to show:

1. that for any \(c\in C\), \(((g\circ f)\circ(f^{-1}\circ g^{-1}))(c) = c\), and
2. that for any \(a\in A\), \(((f^{-1}\circ g^{-1})\circ (g\circ f))(a) = a\).

I leave it for you to verify this analytically.

I hope this helps!

 

[karush]

thanks
much appreciate all that

kinda by myself with this right now class hasn't started yet
gota catch bus now