**Where, => is Implies, ~ is Not, & is AND, = is Logically Equivalent**

Prove this using Precedence rule and Law of statement algebra.

(A => ~B) & (B => A) = ~B

**Where, => is Implies, ~ is Not, & is AND, = is Logically Equivalent**

Prove this using Precedence rule and Law of statement algebra.

Prove this using Precedence rule and Law of statement algebra.

- stapel_eliz
**Posts:**1743**Joined:**Mon Dec 08, 2008 4:22 pm-
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carl89 wrote:(A => ~B) & (B => A) = ~BWhere, => is Implies, ~ is Not, & is AND, = is Logically Equivalent

Prove this using Precedence rule and Law of statement algebra.

Please reply with your efforts so far, your book's version of the "Precedence" rule, and your book's definition of "statement algebra" and what its laws are. Thank you.

PRECEDENCE RULES HIGHEST TO LOWEST: **~, &, V(OR), =>, <=>**

THIS IS WHAT I HAVE SO FAR:

LAW OF ALGEBRA FROM MY BOOK

THIS IS WHAT I HAVE SO FAR:

Code: Select all

`(~A V (~B)) & (~B V A) [ELIMINATE =>]`

[(~A V ~B) & ~B] V [(~A V ~B) & A] [DISTRIBUTION]

[~B & (~A V ~B)] V [A & (~A V ~B)] [COMMUNATIVE]

[(~B & ~A) V (~B & ~B)] V [(A & ~A) V (A & ~B)] [DISTRIBUTIVE]

[(~B & ~A) V (~B & ~B)] V [F V (A & ~B)] [CONTRADICTION]

LAW OF ALGEBRA FROM MY BOOK

- stapel_eliz
**Posts:**1743**Joined:**Mon Dec 08, 2008 4:22 pm-
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carl89 wrote:THIS IS WHAT I HAVE SO FAR:Code: Select all

`(~A V (~B)) & (~B V A) [ELIMINATE =>]`

[(~A V ~B) & ~B] V [(~A V ~B) & A] [DISTRIBUTION]

[~B & (~A V ~B)] V [A & (~A V ~B)] [COMMUNATIVE]

[(~B & ~A) V (~B & ~B)] V [(A & ~A) V (A & ~B)] [DISTRIBUTIVE]

[(~B & ~A) V (~B & ~B)] V [F V (A & ~B)] [CONTRADICTION]

So you've proved the statement to be

Also, by what rule did you replace the "if-then" statements?

Im only trying to prove that Left hand side of the statement is logically equivalent to ~B. I tried the truth table method and it worked just fine.

- stapel_eliz
**Posts:**1743**Joined:**Mon Dec 08, 2008 4:22 pm-
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truth table method is not allowed it was just for confirmation so i have to go to this long way and keep getting stuck at same spot every time.

- stapel_eliz
**Posts:**1743**Joined:**Mon Dec 08, 2008 4:22 pm-
**Contact:**

For interested readers:

Assume B. Then (by "if B, then A") A. Then (by "if A, then not-B") not-B.

Assume not-B. Then (by definition) not-B.

Assume A. Then (by "if A, then not-B") not-B.

Assume not-A. Then (by contrapositive of "if B, then A") not-B.

No matter what you start with on the left-hand side, you end up with the right-hand side.

Assume B. Then (by "if B, then A") A. Then (by "if A, then not-B") not-B.

Assume not-B. Then (by definition) not-B.

Assume A. Then (by "if A, then not-B") not-B.

Assume not-A. Then (by contrapositive of "if B, then A") not-B.

No matter what you start with on the left-hand side, you end up with the right-hand side.