Program: English — B10 · Bahasa Indonesia — B10 · Mathematics Program home · Beranda Program Matematika · Original source — Original author’s reader — Discrete Mathematics: An Open Introduction, 4th edition · Open related page · Open related page Mathematics program — English Program — Bahasa Indonesia Original author’s website ↗ Book contents Original English by Oscar Levin. This unofficial mirror includes local math and search assets and frozen exercise statements. Remote computation, live checking and external websites still need internet access. Use the original website for its live services.
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Contents Index Embed Dark Mode Prev Up Next \(\usepackage{cancel}
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Worksheet Preview Activity
1.
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).
Source-provided solution — read without completing the interaction
Incorrect. Are you sure? Did we say what happens when he
does eat the broccoli, or only what happens when he doesn’t?
Correct. If he got ice cream, he must have eaten the broccoli, because if he didn’t, then he wouldn’t have had ice cream.
Incorrect. Could there have been a reason that Tommy doesn’t get ice cream even if he did eat his broccoli?
Incorrect. 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.
Source-provided solution — read without completing the interaction
Incorrect.
Correct.
Incorrect. Maybe he just really likes giving you tacos. That’s not enough to say he was a liar, is it?
Incorrect.
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.
Source-provided solution — read without completing the interaction
Correct.
Correct.
Correct.
Correct.
Incorrect.
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.
Source-provided solution — read without completing the interaction
Incorrect. 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.
Correct.
Correct. This is equivalent to the original statement (although here “necessary” is used in a logical sense).
Incorrect. 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.
Correct. Another way of saying this is, “I watch a movie only if I eat popcorn.” This is equivalent to the original statement.
Mathematics program — English Program — Bahasa Indonesia Original author’s website ↗ Book contents Program: English — B10 · Bahasa Indonesia — B10 · Mathematics Program home · Beranda Program Matematika · Original source — Original author’s reader — Discrete Mathematics: An Open Introduction, 4th edition · Open related page · Open related page 
