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Answer 1

Answer:i dont know it is correct but see it

Step-by-step explanation:Definition: A Boolean Algebra is a math construct (B,+, . , ‘, 0,1) where B is a non-empty set, + and . are

binary operations in B, ‘ is a unary operation in B, 0 and 1 are special elements of B, such that:

a) + and . are communative: for all x and y in B, x+y=y+x, and x.y=y.x

b) + and . are associative: for all x, y and z in B, x+(y+z)=(x+y)+z, and x.(y.z)=(x.y).z

c) + and . are distributive over one another: x.(y+z)=xy+xz, and x+(y.z)=(x+y).(x+z)

d) Identity laws: 1.x=x.1=x and 0+x=x+0=x for all x in B

e) Complementation laws: x+x’=1 and x.x’=0 for all x in B

Examples:

• (B=set of all propositions, or, and, not, T, F)

• (B=2A

, U, ∩, c

, Φ,A)

Theorem 1: Let (B,+, . , ‘, 0,1) be a Boolean Algebra. Then the following hold:

a) x+x=x and x.x=x for all x in B

b) x+1=1 and 0.x=0 for all x in B

c) x+(xy)=x and x.(x+y)=x for all x and y in B

Proof:

a) x = x+0 Identity laws

= x+xx’ Complementation laws

= (x+x).(x+x’) because + is distributive over .

= (x+x).1 Complementation laws

= x+x Identity laws

x = x.1 Identity laws

= x.(x+x’) Complementation laws

= x.x +x.x’ because + is distributive over .

= x.x+0 Identity laws

= x.x

b) x+1 =x+(x+x’) Complementation laws

= (x+x)+x’ + is associative

= x+x’ using (a)

= 1 Complementation laws

0.x =(x’.x).x Complementation laws

= x’.(x.x) . is associative

= x’.x using (a)

=0 Complementation laws

c) x+(xy) = x.1+x.y Identity laws

=x.(1+y) because + is distributive over .

=x.1 using (b)

Answer 2

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Proof:

a) x = x+0 Identity laws

= x+xx' Complementation laws

= (x+x).(x+x') because + is distributive over.

= (x+x).1 Complementation laws

= x+x Identity laws

x = x.1 Identity laws

= x.(x+x') Complementation laws

= x.x +x.x' because + is distributive over.

= x.x+0 Identity laws

= X.X

b) x+1=x+(x+x') Complementation laws

= (x+x)+x' + is associative

= x+x' using (a)

= 1 Complen ation laws

O.x =(x²x).x Complementation laws

= x²(x.x). is associative

= x'.x using (a)

=O Complementation laws

c) x+(xy) = x.1+x.y Identity laws

=x.(1+y) because + is distributive over.

=x.1 using (b)

• I Hope it's Helpful My Friend Ayan.