show that the three statements are equivalent

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tonyc1970
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Joined: Fri Aug 02, 2013 4:58 pm
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show that the three statements are equivalent

I need to show that "A $\subset$ B" (Let this = 'a'), "A $\cap$ B$^c$= $\emptyset$" (Let this = 'b') and "A$^c$ $\cup$ B = $U$ (universal set)" (Let this = 'c') are equivalent.

a -> b : For A $\subset$ B: Let x $\in$ A and by definition of subset, x $\in$ B, so 'a' is true. For (A $\cap$ B = $\emptyset$): In this case, x $\in$ A and x $\in$ B$^c$ implies that 'b' is true, but it is known that x $\in$ b and this is a contradiction, and A $\cap$ B$^c$ $\neq$ $\emptyset$.

b -> c : For this, I said: By De Morgan's Law, "A $\cap$ B$^c$ = $\emptyset$" is equivalent to (A $\cap$ B$^c$ )$^c$= $\emptyset$$^c$ which is equivalent to A$^c$ $\cup$ B = $U$ which shows that B -> C is true.

c -> a : I am completely stuck on this and need some help.

Any help is appreciated.

Thanks,

Tony

stapel_eliz
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Joined: Mon Dec 08, 2008 4:22 pm
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I need to show that "A $\subset$ B" (Let this = 'a'), "A $\cap$ B$^c$= $\emptyset$" (Let this = 'b') and "A$^c$ $\cup$ B = $U$ (universal set)" (Let this = 'c') are equivalent.
I'm not sure I follow what you're trying to say, so I'll restate:
Given the following statements:

. . . . .Statement X: $A\, \subset\, B$
. . . . .Statement Y: $A\, \cap\, B^c\, =\, \emptyset$
. . . . .Statement Z: $A^c\, \cup\, B\, =\, U$

To prove: Statements X, Y, and Z are equivalent.
And I believe you're asking for help in showing that Statement Z implies Statement X; that is, you're wanting to prove, given that (A-complement)-union-B is "everything", that then A must be a subset of B.

Before going further, kindly please reply with corrections or confirmation. Thank you!

tonyc1970
Posts: 13
Joined: Fri Aug 02, 2013 4:58 pm
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Re: show that the three statements are equivalent

Hi, and thanks for your reply. You are correct in your assumption, and I need to show that statement Z implies statement X.

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
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I would suggest "element chasing". Assume Statement Z, and then pick an element in $A$ (from Statement X). Since $a\, \in\, A$, then $a\, \not\in\, A^c$. Since $U\, =\, A^c\, \cup\, B$ and since necessarily $a\, \in\, U$, then... what can you say about $a$ with respect to $B$?