Prove: (S ∩ T) ∪ U = S ∩ (T ∪ U)

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Homework Help Overview

The discussion revolves around proving the set equality (S ∩ T) ∪ U = S ∩ (T ∪ U), which involves concepts from set theory, specifically unions and intersections.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to use a contrapositive approach but is uncertain about its applicability to proving set equality. Other participants question the validity of this approach and suggest focusing on demonstrating subset relationships instead.

Discussion Status

Participants are exploring different methods to prove the set equality, with some guidance provided on how to approach the proof by showing subset relationships. There is an ongoing dialogue about the correct interpretation of the problem and the appropriate methods to use.

Contextual Notes

There appears to be confusion regarding the nature of the proof required, specifically the distinction between proving implications versus equivalences in set theory.

mutzy188
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Homework Statement



Prove or give a counterexample to each statement.

(S ∩ T) ∪ U = S ∩ (T ∪ U)


The Attempt at a Solution



If I proved by the contrapositive

S (T ∩ U) ≠ (S ∩ T) ∪ U

where would I go from there. How do I find the contrapositive with the unions and intersections?

Thanks
 
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I don't see how that's a contrpositive... contrapositive involves an implication... but this is an equality you need to prove or disprove...
 
(S ∩ T) ∪ U = S ∩ (T ∪ U)

then:

T∪U ∩ S = S ∩ (T ∪ U)

but then where do I go from here
 
mutzy188 said:
(S ∩ T) ∪ U = S ∩ (T ∪ U)

then:

T∪U ∩ S = S ∩ (T ∪ U)

but then where do I go from here

How did you get that?
 
The contrapositive doesn't work because you're not proving implication, but an equivalence of sets.

Choose an arbitrary element contained in the set on one side of the equality, and then show that it's contained in the set on the other side of the equality. That proves the first set is a subset of the set on the other side. Then start on the other side of the equality and follow the same process. That proves equivalence.

That's the gameplan for your proof.
 
Last edited:

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