In Exercise, determine which, if any, of the three statements are equivalent (see Example)...

In Exercise, determine which, if any, of the three statements are equivalent (see Example).

Example

Which Are Equivalent?

Determine which, if any, of the following statements are equivalent. You may use De Morgan’s laws, the fact that ,information from the variations of the conditional, or truth tables.

a) If you leave by 9 A.M., then you will get to your destination on time.

b) You do not leave by 9 a.m. or you will get to your destination on time.

c) It is false that you will get to your destination on time or you did not leave by 9 A.M.

d) If you do not get to your destination on time, then you did not leave by 9 a.m.

Solution Let

p: You leave by 9 a.m.

q: You will get to your destination on time.

In symbolic form, the four statements are

a) p→q.

b) .

c) .

d) ~q→~p.

Which of these statements are equivalent? Earlier in this section, you learned that p →q is equivalent to . Therefore, statements (a) and (b) are equivalent. Statement (d) is the contrapositive of statement (a). Therefore, statement (d) is also equivalent to statement (a) and statement (b). Statements (a), (b), and (d) all have the same truth table (Table).

Table

Now let’s look at statement (c). To determine whether is equivalent to the other statements, we will construct its truth table (Table) and compare the answer column with the answer columns in Table.

Table

None of the three answer columns of the truth table in Table is the same as the answer column of the truth table in Table. Therefore   is not equivalent to any of the other statements. Therefore, only statements (a), (b), and (d) are equivalent to each other.

a) If Fido is our dog’s name, then Rex is not our dog’s name.

---

b) It is false that Fido is our dog’s name and Rex is not our dog’s name.

---

c) Fido is not our dog’s name or Rex is our dog’s name.