1. [SOLVED] Boolean Algebra

Assume that B is a Boolean algebra with operations + and ·.
For each true statement below, give a detailed proof. For each false statement below, write
out its negation, then give a proof of the negation.

(∀a, b ∈ B) (a + b = 1 ↔ b · ā =
ā)
(∀a, b ∈ B) (a · b = 0 b + ā = ā)
(∀a, b ∈ B) (a + b = a + c → b = c)

for the last one, i had the same proof as http://www.mathhelpforum.com/math-he...b-b-c-b-c.html
but it just doesn't seem right to me...
completely stumped on the first two .

thanks!!!

2. Boolean Algebra Proofs

Hello starman_dx
Originally Posted by starman_dx
Assume that B is a Boolean algebra with operations + and ·.
For each true statement below, give a detailed proof. For each false statement below, write
out its negation, then give a proof of the negation.

(∀a, b ∈ B) (a + b = 1 ↔ b · ā =
ā)
(∀a, b ∈ B) (a · b = 0 b + ā = ā)
(∀a, b ∈ B) (a + b = a + c → b = c)

for the last one, i had the same proof as http://www.mathhelpforum.com/math-he...b-b-c-b-c.html
but it just doesn't seem right to me...
completely stumped on the first two .

thanks!!!
Here are the proofs you need. I'll leave you to supply the names of the Laws I'm using at each stage.

(1) $\displaystyle a+b= 1$

$\displaystyle \rightarrow (a+b)\cdot \bar{a} = 1\cdot\bar{a}$

$\displaystyle \rightarrow a\cdot\bar{a} +b\cdot\bar{a} = \bar{a}$

$\displaystyle \rightarrow 0+b\cdot\bar{a} = \bar{a}$

$\displaystyle \rightarrow b\cdot\bar{a} = \bar{a}$

And $\displaystyle b\cdot\bar{a} = \bar{a}$

$\displaystyle \rightarrow b\cdot\bar{a} + a = \bar{a}+a$

$\displaystyle \rightarrow (b+a)\cdot(\bar{a}+a) = 1$

$\displaystyle \rightarrow (a+b).1 = 1$

$\displaystyle \rightarrow a+b = 1$

(2) $\displaystyle a\cdot b=0$

$\displaystyle \rightarrow a\cdot b + \bar{a} = 0 + \bar{a}$

$\displaystyle \rightarrow (a+\bar{a})\cdot(b+\bar{a}) = \bar{a}$

$\displaystyle \rightarrow 1\cdot(b+\bar{a}) = \bar{a}$

$\displaystyle \rightarrow b+\bar{a} = \bar{a}$

And $\displaystyle b+\bar{a} = \bar{a}$

$\displaystyle \rightarrow a\cdot(b+\bar{a}) = a\cdot\bar{a}$

$\displaystyle \rightarrow a\cdot b + a\cdot \bar{a} = 0$

$\displaystyle \rightarrow a\cdot b + 0 = 0$

$\displaystyle \rightarrow a\cdot b = 0$

(3) is not true. The negation is $\displaystyle \exists a, b, c \in B, (a+b = a+c) \wedge (b \ne c)$

As a counter-example, consider sets $\displaystyle A = \{1\}, B = \{2\}$ and $\displaystyle C = \{1, 2\}$. Then $\displaystyle A \cup B = A \cup C$, but $\displaystyle B \ne C$.