Prove or disprove: If Ā⊆B then A∪B=U.?

I need the proof or a counter example.

Thanks!

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  • 10 years ago
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    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

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