show that the three statements are equivalent  TOPIC_SOLVED

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show that the three statements are equivalent

Postby tonyc1970 on Fri Dec 13, 2013 10:43 pm

I need to show that "A B" (Let this = 'a'), "A B= " (Let this = 'b') and "A B = (universal set)" (Let this = 'c') are equivalent.

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

b -> c : For this, I said: By De Morgan's Law, "A B = " is equivalent to (A B )= which is equivalent to A B = 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
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Postby stapel_eliz on Sat Dec 14, 2013 1:53 am

tonyc1970 wrote:I need to show that "A B" (Let this = 'a'), "A B= " (Let this = 'b') and "A B = (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:
. . . . .Statement Y:
. . . . .Statement Z:

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! :wink:
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Re: show that the three statements are equivalent

Postby tonyc1970 on Sat Dec 14, 2013 6:11 pm

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

Postby stapel_eliz on Sun Dec 15, 2013 2:52 am

I would suggest "element chasing". Assume Statement Z, and then pick an element in (from Statement X). Since , then . Since and since necessarily , then... what can you say about with respect to ?
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