Question

I'm not sure why I'm getting this wrong. For the sets A= {1, 2, . . . , 10} and B = {2, 4, 6, 9, 12, 25}, consider the statements P: A is a subset of B. Q: |A-B| = 6. Is P V Q true or false? I said it's false because neither is true. A is not a subset of B, and the difference of the cardinalities of the sets is not 6. However, my book said it's true. I'm confused. :(

Answer 1

What is the set A - B? It is
\[A - B = \{ a \in A \ | \ a \notin B \} = \{ 1, 3, 5, 7, 8, 10 \}\]

Answer 2

I'm sure you now see your problem.

Answer 3

P: A is a subset of B

This is false since 1 (the first element in set A) is NOT in set B

---------------------------

Q: |A-B| = 6

To find A - B, start with A and erase all elements found in set B

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A-B = {1, 3, 5, 7, 8, 10}


Now count the number of elements. There are 6 elements here, so


|A - B| = 6


So Q: |A-B| = 6 is true


Therefore, P v Q is true

Answer 4

Thanks Jim. I didn't understand how that cardinality was working there. It makes sense now!

Answer 5

Thanks James, yes. Awesome.

Answer 6

In the notation of your last question

||A - B|| = ||A|| - ||A n B||
= 10 - 4

and the critical thing is there are other elements of B which are not in A; but they play no role here. That's where you were getting caught up.

Answer 7

I thought it was the difference of the cardinalities. My professor and book did not explain this as far as I can tell.

Answer 8

As this example, show us, no.

Here's another example

A = { 2 }
B = { prime numbers less than 1,000,000,000 }

Clearly the cardinality of A - B is zero, but it is not the cardinality of A (which is 1) less the cardinality of B (which is something quite a bit bigger than 1)

Answer 9

|A - B| = the number of elements in set A - B

This is the same as saying that

|A| = the number of elements in set A

Answer 10

Thanks everyone, it is much clearer now. Math logic is new to me.