Consider the statement, “If Tommy doesn’t eat his broccoli, then he will not get any ice cream.” Which of the following statements mean the same thing (i.e., will be true in the same situations)? Select all that apply.
If Tommy does eat his broccoli, then he will get ice cream.
Are you sure? Did we say what happens when he does eat the broccoli, or only what happens when he doesn’t?
If Tommy gets ice cream, then he ate his broccoli.
If he got ice cream, he must have eaten the broccoli, because if he didn’t, then he wouldn’t have had ice cream.
If Tommy doesn’t get ice cream, then he didn’t eat his broccoli.
Could there have been a reason that Tommy doesn’t get ice cream even if he did eat his broccoli?
Tommy ate his broccoli and still didn’t get any ice cream.
This is the opposite of the original statement (it is false precisely when the original statement is true).
2.
Suppose that your shady uncle offers you the following deal: If you loan him your car, then he will bring you tacos. In which of the following situations would it be fair to say that your uncle is a liar (i.e., that his statement was false)? Select all that apply.
You loan him your car. He brings you tacos.
You loan him your car. He never buys you tacos.
You don’t loan him your car. He still brings you tacos.
Maybe he just really likes giving you tacos. That’s not enough to say he was a liar, is it?
You don’t loan him your car. He never brings you tacos.
3.
Consider the sentence, “If \(x \ge 10\text{,}\) then \(x^2 \ge 25\text{.}\)” This sentence becomes a statement when we replace \(x\) by a value, or “capture” the \(x\) in the scope of a quantifier. Which of the following claims are true (select all that apply)?
If we replace \(x\) by \(15\text{,}\) then the resulting statement is true. (Note, \(15^2 = 225\text{.}\))
If we replace \(x\) by \(3\text{,}\) then the resulting statement is true.
If we replace \(x\) by \(6\text{,}\) then the resulting statement is true.
The universal generalization (“for all \(x\text{,}\) if \(x \ge 10\) the \(x^2 \ge 25\)”) is true.
There is a number we could replace \(x\) with that makes the statement false.
4.
Consider the statement, “If I see a movie, then I eat popcorn” (which happens to be true). Based solely on your intuition of English, which of the following statements mean the same thing? Select all that apply.
If I eat popcorn, then I see a movie.
This is not equivalent to the original statement. Maybe I also eat popcorn when I watch TV? In that case, the original statement would be true, but this one would be false.
If I don’t eat popcorn, then I don’t see a movie.
Correct.
It is necessary that I eat popcorn when I see a movie.
This is equivalent to the original statement (although here “necessary” is used in a logical sense).
To see a movie, it is sufficient for me to eat popcorn.
Just because I eat popcorn, doesn’t mean I see a movie. I might eat popcorn in other situations. So this is not equivalent to the original statement.
I only watch a movie if I eat popcorn.
Another way of saying this is, “I watch a movie only if I eat popcorn.” This is equivalent to the original statement.
