Kenneth Rosen Edition 6th Exercise 2.2 Question 15 (Page No. 130)

Question

If a,b be elements of a Boolean algebra then how to show that (a∨b)' = a' ∧ b'.

please explain step by step in easiest way possible

Answer 1

You can make use of complement postulates AvA'=1 and A∧A'=0 to prove equality of two given boolean expressions.
To prove (a∨b)' = a' ∧ b', you will have to prove
1. (a∨b)'' v (a' ∧ b') = 1
 =>(a∨b)'' v (a' ∧ b') = (a∨b) v (a' ∧ b')
                                = (avb)va' ∧ (avb)vb' [Distributive law]
                                = (ava')vb ∧ av(bvb') [Associative law]

                                = (1vb) ∧ (av1) [Complement law]
                                = 1  ∧ 1 [Nullity law]
                                = 1 [Identity law]
2. (a∨b)'' ∧ (a' ∧ b') = 0
=>(a∨b)'' ∧ (a' ∧ b') = (a∨b) ∧ (a' ∧ b')
                               = a∧(a'∧b') ∨ b∧(a'∧b')   [Distributive law]
                               = (a∧a')∧b' ∨ (b∧b')∧a'  [Associative law]
                               = (0∧b') ∨ (0∧a')  [Complement law]
                               = 0 v 0  [Nullity law]
                               = 0 [Identiy law]

Another easy way to prove will be to construct truth table for L.H.S and R.H.S and then check for equality.

Comments

CAN U ALSO POST THE OTHER WAY USING TRUTH TABLE.

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You can also use venn diagram concept to prove it

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ok i have added the truth table too.

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Thanks Srinath! :)

Answer 2

we can say two formulas are equal if left side and write side have the same truth values in all the cases

SO lets see saying it is not equal 

so take A=(aᐯb)' 

B=a' ^ b'

so to Prove that we can assign A=True and B=False in some case it is enough to show two statements are not equal

So B=False means either a or b are False 

So take those values of a and b and check if we are getting a= True value for A which we will not get so it is a valid statement