SUNY Geneseo Department of Mathematics
Monday, February 4
Math 239 01
Spring 2019
Prof. Doug Baldwin
Section 2.1
Classify the following as true or false, and say why:
Let G be the statement “Geneseo won the 1950 Superbowl” and E be the statement “there are pink elephants dancing on the classroom tables.” Write symbolic forms for (at least) two interpretations of the last example above (it is not the case that Geneseo won the 1950 superbowl and there are pink elephants dancing on the tables).
The negation applies only to Geneseo winning the Superbowl: ¬G ∧ E
The negation applies to the whole “and”: ¬( G ∧ E )
The negation distributes over the “and”: ¬G ∧ ¬E
But are the second and third expressions really different? In fact they are: consider any case in which one of G and E is true and the other false. Then in the second expression G ∧ E is false, so its negation is true. But in the third expression, one of ¬G and ¬E is true and the other false, so the whole “and” is false. Apparently “not” doesn’t distribute over “and.” But it would be nice to have a more systematic way to explore such differences...
Write a truth table for the statement (P ∧ ¬ Q) ∨ R, where P, Q, and R are arbitrary statements.
| P | Q | R | ¬Q | (P∧ ¬ Q) | (P ∧ ¬ Q) ∨ R, |
|---|---|---|---|---|---|
| T | T | T | F | F | T |
| T | T | F | F | F | F |
| T | F | T | T | T | T |
| T | F | F | T | T | T |
| F | T | T | F | F | T |
| F | T | F | F | F | F |
| F | F | T | T | F | T |
| F | F | F | T | F | F |
Write truth tables for ¬(P ∨ Q) and ¬P ∧ ¬Q.
| P | Q | P ∨ Q | ¬(P ∨ Q) |
|---|---|---|---|
| T | T | T | F |
| T | F | T | F |
| F | T | T | F |
| F | F | F | T |
| P | Q | ¬P | ¬Q | ¬P ∧ ¬Q |
|---|---|---|---|---|
| T | T | F | F | F |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |
Since the last columns of these two tables are identical, it seems that ¬G ∧ ¬E from the symbolic notation discussion is actually equivalent to ¬( G ∨ E ). There is an algebra of logical (Boolean) expressions, it just doesn’t always behave the way you expect. To learn more...
Equivalence of compound statements and Boolean algebra.
Read section 2.2