Determine which of the following pairs of statements forms are logically equivalent. Justify your answer using truth tables and include a few words of explanation. Read "t" to be a tautology and "c" to be a contradiction:
26. (p^q)v(p^r)and(pvq)^r
28. (rvp)^((~rv(p^q))^(rvq))and p^q
To determine which of the pairs of statements are logically equivalent, we can create truth tables for each statement and compare the columns for the resulting truth values. If two statements have identical columns, they are logically equivalent.
Let's analyze each pair of statements one by one:
26. (p^q)v(p^r) and (pvq)^r
Truth table for (p^q)v(p^r):
| p | q | r | p^q | p^r | (p^q)v(p^r) |
|---|---|---|-----|-----|-------------|
| T | T | T | T | T | T |
| T | T | F | T | F | T |
| T | F | T | F | T | T |
| T | F | F | F | F | F |
| F | T | T | F | F | F |
| F | T | F | F | F | F |
| F | F | T | F | F | F |
| F | F | F | F | F | F |
Truth table for (pvq)^r:
| p | q | r | pvq | (pvq)^r |
|---|---|---|-----|---------|
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | T | T |
| T | F | F | T | F |
| F | T | T | T | T |
| F | T | F | T | F |
| F | F | T | F | T |
| F | F | F | F | F |
By comparing the columns for the resulting truth values, we can conclude that the statements (p^q)v(p^r) and (pvq)^r are not logically equivalent. They have different resulting truth values in rows 2 (when p and q are both true and r is false) and row 6 (when p and q are both false and r is true).
28. (rvp)^((~rv(p^q))^(rvq)) and p^q
Truth table for (rvp)^((~rv(p^q))^(rvq)):
| p | q | r | rvp | ~rv(p^q) | rvq | (~rv(p^q))^(rvq) | (rvp)^((~rv(p^q))^(rvq)) |
|---|---|---|-----|----------|-----|------------------|------------------------|
| T | T | T | T | F | T | F | F |
| T | T | F | T | T | F | F | F |
| T | F | T | T | F | F | F | F |
| T | F | F | T | T | F | T | T |
| F | T | T | F | F | T | F | F |
| F | T | F | F | F | F | F | F |
| F | F | T | F | F | T | F | F |
| F | F | F | F | F | F | F | F |
Truth table for p^q:
| p | q | p^q |
|---|---|-----|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
By comparing the columns for the resulting truth values, we can conclude that the statements (rvp)^((~rv(p^q))^(rvq)) and p^q are not logically equivalent. They have different resulting truth values in rows 2 (when p and q are both true) and rows 4 and 5 (when p and/or q are false).
In conclusion, none of the pairs of statements are logically equivalent in this case.