# Propositional Logic questions (Conditionals).

I am uncertain if I did the following questions correctly, and I will appreciate it very much if anyone can help point out any errors in my procedure.

Question 1: Let $u$ mean "the Racket identifier may be used" and $d$ mean "the Racket identifier has been defined". Which of the following means "The Racket identifier may not be used if it has not been defined."?

A)$\lnot u \rightarrow \lnot d$

B)$\lnot d \rightarrow \lnot u$

C) None of the above

I think the anwser should be B). At first, I thought of A) as the correct answer, but on a second thought, in the statement "The Racket identifier may not be used if it has not been defined." did not use only if. So, this statement is equivalent to "if it has not been defined, then the Racket identifier may not be used.

Question 2: Let $t$ mean "Mary-Kate and Ashley are identical twins." Let $m$ mean "Mary-Kate has blue eyes." Let $a$ mean "Ashley has blue eyes." Which of the following means "If Mary-Kate and Ashley are identical twins, then either they both have blue eyes or neither has blue eyes."

A) $t \rightarrow (m \land a)$

B) $t \leftrightarrow (m \leftrightarrow a)$

C) $t \rightarrow (m \leftrightarrow a)$

D) $t \leftrightarrow (m \rightarrow a)$

E) None of the above.

I believe the answer should be E), since the statement "If Mary-Kate and Ashley are identical twins, then either they both have blue eyes or neither has blue eyes." translates to the conditional: $t \rightarrow ((m \land a) \lor \lnot(m \lor a))$, and the consequent does not correspond to any conditional/biconditional statements in A) through D). So E) may be the answer.

Question 3: Which of the following could be the result of applying the negation law to the statement:$p \land (a \lor b) \land \lnot (a \lor b)$:

A) $p \lor \lnot (a \lor b)$

B) $p \land T$

C) $p \land (a \lor b)$

D) $p \land F$

E) None of the above

According to my textbook, the negation law states that $p \lor \lnot p \equiv T$ and $p \land \lnot p \equiv F$. So I can use first use associative law to get $p \land ((a \lor b) \land \lnot (a \lor b))$ then apply negation law to get $p \land F$. So the answer is D)

Also, what does the F and T mean in the negation law. I mean $p \land \lnot p$, does this is a tautological falsehood (contradiction) right?

• $F$ is False and $T$ is True; thus, the negation of $F$ is $T$ and the same for the other case. Sep 11, 2018 at 19:24

I think the anwser should be B). At first, I thought of A) as the correct answer, but on a second thought, in the statement "The Racket identifier may not be used if it has not been defined." did not use only if. So, this statement is equivalent to "if it has not been defined, then the Racket identifier may not be used.

Yes. $\neg d\to\neg u$ says "Not $u$ if not $d$", and also "If not $d$, then not $u$."

I believe the answer should be E), since the statement "If Mary-Kate and Ashley are identical twins, then either they both have blue eyes or neither has blue eyes." translates to the conditional: $t→((m∧a)∨¬(m∨a))$, and the consequent does not correspond to any conditional/biconditional statements in A) through D). So E) may be the answer.

No. The consequent is saying that the truth values for $m$ and $a$ are equal, which is a bicondition. $${\quad(m\wedge a)\vee\neg(m\vee a) \\\equiv (m\wedge a)\vee(\neg m\wedge \neg a)\\\equiv (m\vee\neg a)\wedge(\neg m\vee a)\\\equiv (a\to m)\wedge(m\to a)\\\equiv m\leftrightarrow a}$$

According to my textbook, the negation law states that $p∨¬p≡T$ and $p∧¬p≡F$. So I can use first use associative law to get $p∧((a∨b)∧¬(a∨b))$ then apply negation law to get $p∧F$. So the answer is D)

Indeed, that is so.

Also, what does the F and T mean in the negation law. I mean $p∧¬p$, does this is a tautological falsehood (contradiction) right?

T : true, F : false. $\phi\vee\neg\phi$ is true, $\phi\wedge\neg\phi$ is false.

Indeed since they are true and false reguardless of the value of $\phi$, they are a tautology and contradiction respectively.