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

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  • Thread starter karush
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In summary, we discussed a proof for the set identity $A\cap(B/C)=(A\cap B)/(A\cap C)$ and showed that for invertible mappings $f:A\rightarrow B$ and $g:B\rightarrow C$, $(g\circ f)^{-1}= f^{-1}\circ g^{-1}$. You will need to be more formal with your proof for the set identity and verify analytically for the invertible mappings. Good luck with your class!
  • #1
karush
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$\tiny{411.1.3.15}$

$\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
 
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  • #2
karush said:
$\tiny{411.1.3.15}$

$\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!
 
  • #3
thanks
much appreciate all that

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

FAQ: 411.1.3.15 Prove A\cap(B/C)=(A\cap B)/(A\cap C)

1. What does "411.1.3.15" in the equation represent?

The numbers in the equation are just a label or identifier. They do not have any specific meaning in relation to the equation itself.

2. What does the symbol "/" mean in the equation?

The "/" symbol in this equation represents the set difference operation. It is used to indicate that the elements in the following set are not present in the set before the "/" symbol.

3. How do you prove the equation "A∩(B/C) = (A∩B)/(A∩C)"?

To prove this equation, we need to show that both sides of the equation contain the same elements. We can do this by showing that an element belongs to one side if and only if it belongs to the other side. This can be done by using the definition of set intersection and set difference operations.

4. What does "A∩(B/C)" mean in the equation?

"A∩(B/C)" represents the intersection of set A with the set resulting from the difference of set B and C. This means that the resulting set contains all elements that are present in both set A and the set resulting from removing elements in C from set B.

5. Why is this equation important in mathematics?

This equation is important in mathematics because it is a fundamental property of sets and helps us understand the relationships between different sets. It also has practical applications in solving problems related to set operations and proving theorems in various branches of mathematics.

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