Best Answer:
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

Source(s):

Asker's rating