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) Fitz and the Tantrums will not go on tour if and only if James King does not play the saxophone.
b) It is false that Fitz and the Tantrums will go on tour if and only if James King does not play the saxophone.
c) If Fitz and the Tantrums go on tour, then James King plays saxophone.
We need at least 10 more requests to produce the solution.
0 / 10 have requested this problem solution
The more requests, the faster the answer.