Test by Bikram | Mock GATE | Test 2 | Question: 47

Question

Which of the following statements is/are true?

1. $(a\rightarrow b)$ always equals $\bar{a} +b.$ 
2. $\bar a+b  +a \bar b $ is a tautology.
3. $\left (a\rightarrow b \right).\left(a \bar b \right )$  is satisfiable.

• A. III only
• B. I and II only
• C. II and III only
• D. I, II, and III

Comments

what Is star here ??

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* is multiplication

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C is satisfiable,just put A=false,B=false/true.

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Kabi

C is not satisfiable, it is a contradiction

when u put A = false, the first term (A→ B) is always true but then(ab') is always false because it is (a∧b'), So entire expression can never be true.

 I and II only is a true

Answer 1

1) a -->b = a' v b = a'+b (Always True)

2) (~a+b) + (a*~b) = a'+b+ab' = a'+ab'+b = a'+b'+b = a'+1 = 1 (Tautology)   Here a'+ab' = a'+b'

3) (a-->b)*(a*~b) = (a'+b)*ab' = a'ab'+bab' = 0 (Falsifiable)

Hence 1) and 2) are true so correct ans is B.