**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:**1628**Joined:**Mon Dec 08, 2008 4:22 pm-
**Contact:**

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.(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.

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

THIS IS WHAT I HAVE SO FAR:

[/size]

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:**1628**Joined:**Mon Dec 08, 2008 4:22 pm-
**Contact:**

So you've proved the statement to beTHIS IS WHAT I HAVE SO FAR:

[/size]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]`

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:**1628**Joined:**Mon Dec 08, 2008 4:22 pm-
**Contact:**

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:**1628**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.