Hey this is Presh Talwalkar. Which of the statements, if any, are false?
This is a list of ten statements. The nth statement in this list claims that exactly
n of the statements are false. Can you figure it out? While you're working on
that problem, also try and figure out this variation.
It's the same list of 10 items except the word "exactly" is replaced by the words "at least".
Give these problems a try, and when you're ready keep watching the video for the solution.
We'll get started with the list that the nth statement says that exactly n statements are false.
Consider the tenth statement in this list.
If statement 10 is true, that implies all of the statements in the list are false, including statement 10.
This is a contradiction which means statement 10 must be false.
So we know at least one of the statements in this list is false.
How many of the statements can be true at the same time?
The statements in the list are mutually exclusive; you cannot have exactly x statements be true and exactly y statements be true for x not equal to y.
This implies at most one statement can be true.
If one statement is true that would mean nine statements
are false.
So our candidate is we have nine false statements, one true statement.
This corresponds to the claim of statement nine. If statements one through
eight, and 10 are false and statement nine is true, that actually works out logically.
So that's our answer. Statement nine is true and statements one through eight, and statement 10 are false.
We have nine false statements and one true statement.
Now let's solve the variation where the nth statement says at least n
statements are false. We'll proceed similarly. We'll consider the 10th statement.
If this is true, that means all statements in the list are false
including statement 10.
This again is a contradiction.
So we know that statement 10 is false , and that means at least one statement must be false.
So if at least one statement is false,
we know that the first statement has to be true.
Now we proceed to the next statement. If statement nine is true, that would mean
statements 2 through 9 have to be false because we already know statement one is
true. Therefore statement 9 has to be false, and that implies at least two
statements are false, statement nine and ten are false
so we know that statement two is true. We can similarly proceed to show that
statements eight, seven, and six are false while three, four, and, five must be true.
So what we end up with is we have five possibly true statements and five
possibly false statements. And this turns out to logically work, so statements
1 through 5 are true, and statements 6 through 10 are false.
Did you figure it out?
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