Yahoo Answers: Answers and Comments for Prove or disprove: If Ā⊆B then A∪B=U.? [Mathematics]
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From ovdoman9
enUS
Mon, 05 Apr 2010 17:39:20 +0000
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Yahoo Answers: Answers and Comments for Prove or disprove: If Ā⊆B then A∪B=U.? [Mathematics]
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From Ex Falso Quodlibet: Hi
If Ā ⊆ B, then everything that is not in...
https://answers.yahoo.com/question/index?qid=20100405173920AAiw8p0
https://answers.yahoo.com/question/index?qid=20100405173920AAiw8p0
Tue, 06 Apr 2010 02:09:52 +0000
Hi
If Ā ⊆ B, then everything that is not in A is in B. Everything that is in B is in A ∪ B. Thus, if Ā ⊆ B, then everything that is not in A is in A ∪ B.
Everything that is in A is in A ∪ B.
Hence, since everything is either in or not in A, everything is in A ∪ B so long as Ā ⊆ B. To say that everything is in A ∪ B is just to say that A ∪ B = U.
We can prove this formally as follows. Let 'Ax' read 'x belongs to A' and 'Bx' read 'x belongs to B'. Then we can symbolise Ā ⊆ B as (Vx)(~Ax > Bx) and A ∪ B = U as (Vx)(Ax v Bx).
(Vx)(~Ax > Bx) entails (in fact, is logically equivalent to) (Vx)(Ax v Bx), so Ā ⊆ B implies that A ∪ B = U.
(1) 1. (Vx)(~Ax > Bx) Premise
(1) 2. ~Ac > Bc 1 UE
(3) 3. ~(Ac v Bc) Assumption
(4) 4. Ac Assumption
(4) 5. Ac v Bc 4 vI
() 6. Ac > (Ac v Bc) 4,5 CP
(3) 7. ~Ac 3,6 MT
(1,3) 8. Bc 2,7 MP
(1,3) 9. Ac v Bc 8 vI
(1,3) 10. (Ac v Bc) & ~(Ac v Bc) 9,3 &I
(1) 11. ~~(Ac v Bc) 3,10 RAA
(1) 12. Ac v Bc 11 DNE
(1) 13. (Vx)(Ax v Bx) 12 UI