Boolean Algebra

A non-empty set B under two binary operations denoted by '+' and '.' and one unary operation, ''' is said to be a Boolean Algebra, if it satisfies the following properties.

$B_{1}$ : For all x, y $\in$ B, (i) x+y = y+x (ii) x-y = y.x

$B_{2}$ : For all x, y, z $\in$ B (i) x+(y.z) = (x+y) . (x+z) (ii) x.(y+z) = (x.y) + (x.z)

$B_{3}$ : For all x $\in$ B (i) x+0=x (ii) x.1 = x

'0' is called zero element and '1' is called unit element.

$B_{4}$ : For all x $\in$ B, there exist x' $\in$ B such that (i) x+x'= 1 and (ii) x.x'= 0

Results

(1) (x+y) + z = x+ (y+z) and (x.y) . z = x.(y.z)

(2) a+a = a and a . a = a

(3) a+1 = 1 and a . 0=
0

(4) ${(a')}^{'}$ = a

(5) ${(a . b)}^{'}$ = a' + b' and ${(a + b)}^{'}$ = a' . b'

(6) x + (x.y) = x and x.(x+y) = x

(7) 0' = 1 and 1' = 0

(8) x+ y = 1 and x.y= 0 $\Rightarrow$ y = x'

Dual Statement: A dual statement is obtained by interchanging '+' and '.' and '1' and '0'.

Truth Table

p q p+q p.q $\sim p$ p $\to q$ p $\leftrightarrow q$

T T T T F T T

T F T F F F F

F T T F T T F

F F F F T T T

p	q	p+q	p.q	$\sim p$	p $\to q$	p $\leftrightarrow q$
T	T	T	T	F	T	T
T	F	T	F	F	F	F
F	T	T	F	T	T	F
F	F	F	F	T	T	T