Propositional Logic

Question

Ans. B

Comments

How i approached:-

(a or b) and (not a or c) and not b and not c ----> (a+b).(!a + c).!b.!c  ----> In boolean algebra.

(a+b)(!a!b!c) = F. Hence contradiction.

Can anybody explain me what is contingency ?

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A proposition that is neither a tautology nor a contradiction is called a contingency. 
For example (P  Q) is a contingency. 
Source: https://www.cs.odu.edu/~cs381/cs381content/logic/prop_logic/tautology/tautology.html

How to check?

Check for tautology, if not then contradiction if not then its contingency.

Answer 1

can be done in two ways : 

1) if u replace union by + and intersection by *.

=(a+b)(a'+c)(b+c)'    (basic digital theorems)

=(ac+a'b+bc)(b'c') = 0 (contradiction)

2)((avb) ∧ (a'vc))∧ (bvc)'

((bVa) ∧ (a'vc))∧ (bvc)'   ( As a--> b <==> a' v b)

((b'-->a)∧  (a-->c) ) ∧ (bvc)'

(b'-->c) ∧ (bvc)'     (transposition rule)

(b v c) ∧ (bvc)' = 0 (negation rule P ∧ P' =0)

Hence a contradiction.

Answer 2

Yes, because it can be written as (a+b)(a'+c)(b+c)'
= (a+b)(a'+c)(b'c')            by DeMorgan's Law
= (ab'c')(a'+c)                   by Commutativity
= 0

Hence a contradiction. ie B)

Answer 3

This can also prove by using assuming that  a is having truth value than, using the equivalence formula we can prove that weather is  tautology or contradiction 

 

{(a+b)*('a+c)*'(b+c)}

a=T;