Identify which of the following are tautologies….

Apparently a. and d. are tautologies, but I cannot understand why so I do not know how to prove either.

(8 pts) 7. Identify which of the following (if any) are tautologies regardless of P, Q and the universe of discourse for x. If the statement is a tautology prove it. Be sure to state all laws or rules of inference that you use in your proof. b. axPOx) A3xQ(x) 3x(P(x), A, Q(x))

## Expert Answer

a)

LHS: this is true only when there exist x (let it be x1) for which both P(x) and Q(x) holds true (i.e. P(x1) = true, and Q(x1) = true)

RHS: If LHS is true, we cannot make LHS false (since we have atleast one x, for which both P() and Q() are true. So obhiously there exist a x for which P() is true and there exist a x for which Q() is true)

So this is Tautology

b)

LHS: This is true when there exist a x (let it be x1) for which P(x) is true and there exist a x,may be different, (let it be x2) for which Q(x) is true.

RHS: If LHS is true, we cannot be sure RHS is true, because RHS demands there exist x (let it be x1) for which both P(x) and Q(x) holds true (i.e. P(x1) = true, and Q(x1) = true)

so this is not Tautology

c)

LHS: is True when for all x either P(x) is true or Q(x) is true.

RHS: If LHS is true we cannot comment on RHS, since RHS demands for all x P(x) is true or for all x Q(x) is true. (difference is in LHS it states that we can have x1 such that P(x1) is false, but Q(x1) should be true or vice-versa. But RHS says either P(x) is true for all X or Q(x) is true for all X)

so this is not Tautology

d)

LHS:This is true if for all x P(x) is true or for all x Q(x) is true.

RHS:True value of LHS confirms RHS is True because RHS demands for all x either P(x) is true or Q(x) is true.

so this is Tautology

I hope you like the explanation provided. If you have and doubt please feel free to comment below. I shall be glad yo help you