Questions?
Compound Statements and Connectives
Section 2.1
Meanings of Connectives
Classify the following as true or false, and say why:
- Today is Monday and there are pink elephants dancing on the classroom tables. False, since 1 part of the “and” is false
- Today is Monday or there are pink elephants dancing on the classroom tables. True, this is an “or” with 1 part true
- Today is Monday or π is between 3 and 4. True, an “or” with both parts true
- It’s not the case that Geneseo won the 1950 superbowl. True, Geneseo didn’t win
- Today is not Monday. False
- There are pink elephants dancing on the classroom tables only if today is Monday. True, this is equivalent to “if pink elephants are dancing on the classroom tables then today is Monday,” which is a conditional with a false hypothesis.
- It is not the case that Geneseo won the 1950 superbowl and there are pink elephants dancing on the tables. One reading has this false because it’s an “and” whose first part is negated and whose second part is the false statement “there are pink elephants dancing on the tables.” But other readings can apply the negation to other parts, and give a true statement. Hmm, we need a way to write compound statements that’s more precise than English...
Symbolic Notation
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...
Truth Tables
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...
Next
Equivalence of compound statements and Boolean algebra.
Read section 2.2