Predicate Logic

Question

Comments

B seems correct

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Can you please explain option C.Thanks in advance

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LHS implies RHS

LHS is true when A(x) and B(x) both are true for all x  or when A(x) and B(x) both not  true for all x 

RHS says wherever A(x) is true B(x) also true

Counterexample is given in answer

Answer 1

First Statement says IF for all x either A(x) or B(x) be true then either A(x) true for all x or B(x) true for all x 

Below example serves as counter example

Let's take three values x = a,b,c such that A(a) = False A(b) = A(c) = True and B(a) = True B(b) = B(c) = False

Second Statement says IF for all x A(x) ==> B(x) [i.e. wherever A(x) is true, B(x) also true]  then if for all x  A(x) true implies B(x)  true for all x      Correct

Third Statement Counter example

Let's take three values x = a,b,c such that A(a) = False A(b) = A(c) = True and B(a) = True B(b) = B(c) = False