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2 set theory problems

  • Thread starter sbc824
  • Start date
  • #1
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Homework Statement



show S1 U S2 = (S1' ∩ S2')'

The Attempt at a Solution



I'm pretty sure I have this right or I'm close

Let x ∈ S1 U S2
x ∈ S1 or x ∈ S2
Since x ∈ S1 or S2, then x ∉ S1' and S2'
If x ∉ S1' and S2', then x ∈ (S1' and S2')'
Therefore, S1 U S2 = (S1' ∩ S2')'

Homework Statement



show S1 U S2 - (S1 ∩ S2') = S2

The Attempt at a Solution



I have not attempted this as I'm not sure how to start this one...any help would be appreciated.
 
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Answers and Replies

  • #2
SammyS
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Homework Statement



show S1 U S2 = (S1' ∩ S2')'

The Attempt at a Solution



I'm pretty sure I have this right or I'm close

Let x ∈ S1 U S2
x ∈ S1 or x ∈ S2
Since x ∈ S1 or x ∈ S2, then x ∉ S1' and x ∉ S2' This is not correct.

If x ∈ S1, then it does not need to be in S2. If it's not in S2, then x ∈ S2'.


If x ∉ S1' and x ∉ S2', then x ∈ (S1' and S2')'
Therefore, S1 U S2 = (S1' ∩ S2')'

Homework Statement



show S1 U S2 - (S1 ∩ S2') = S2

The Attempt at a Solution



I have not attempted this as I'm not sure how to start this one...any help would be appreciated.
You don't have the first part right.
 
  • #3
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0
You don't have the first part right.
wow silly mistake thanks...any starting hints for 2? I can easily visualize it with a diagram...but I'm rusty with set notation.
 
  • #4
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,302
998
wow silly mistake thanks...any starting hints for 2? I can easily visualize it with a diagram...but I'm rusty with set notation.
Another problem with your solution to part 1 is that you have only shown that S1 U S2 ⊆ (S1' ∩ S2')' (that is, if you have truly corrected your proof). To show equality, you also need to show that S1 U S2 ⊇ (S1' ∩ S2')' .
 
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