DSCI 220, 2025 W1
September 12, 2025
Given \(p\rightarrow q\), we define the following 3 terms:
________________ is \(\neg q\rightarrow \neg p\)
________________ is \(q\rightarrow p\)
________________ is \(\neg p\rightarrow \neg q\)
Describe all the logical equivalences among the 4 statements:
True| # | Statement p→q | p→q | q→p |
|---|---|---|---|
| 1 | To be empty, a string must have length 0. | ☐ | ☐ |
| 2 | Playing in the NHL entails being a professional hockey player. | ☐ | ☐ |
| 3 | If an animal is a mammal, then it is a cat. | ☐ | ☐ |
| 4 | Riding the 99 B-Line counts as public transit. | ☐ | ☐ |
| 5 | If a dish is spicy, then it is Indian cuisine. | ☐ | ☐ |
| 6 | If it’s a kitten, then it’s a cat. | ☐ | ☐ |
| 7 | No song appears on the Billboard Hot 100 without reaching #1. | ☐ | ☐ |
| 8 | If I live in Totem Park residence, then I live in campus housing. | ☐ | ☐ |
| 9 | If I eat breakfast, then I ace the quiz. | ☐ | ☐ |
| 10 | If a movie is a superhero film, then it is an MCU movie. | ☐ | ☐ |
| 11 | If I’m an only child, then I have zero siblings. | ☐ | ☐ |
| 12 | If I’m in British Columbia, then I’m in Vancouver. | ☐ | ☐ |
We have a special operator for those implications whose converses are also True:
Some phrases describing this relationship:
Which of the expressions on the previous page are biconditionals?
Consider the predicate:
Is this a proposition? ____. We add a “quantifier”:
Now is it a proposition? ____. We scope the variable:
Two classic quantifiers:
In pairs, complete the worksheet!
What did you discover about negation?
Suppose someone says “all comedians are funny,” and you disagree, how would you refute the statement?
Your immediate response: __________
This evidence of refutation is called a _______________.
Another way:
Applies specifically to \(\neg\forall\)
Suppose someone says “there is a funny comedian,” and you disagree, how could they justify their statement?
Their immediate response: __________
This evidence of justification is called a _______________.
Applies specifically to \(\exists\)
Aside: Why not \(\exists x, C(x)\rightarrow F(x)\)?
It will not surprise you that we can use multiple quantifiers:
In pairs, complete the second worksheet.
Quantifiers – “For all” and “There exists”
Counterexample and Witness
Negating quantifiers
Nested Quantifiers