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October 28th, 2015, 09:22 PM   #1
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Infinite sets question

Michael asserts that if A and B are infinite sets, A n B must be an infinite set. Is he correct? Explain.

My attempt.

Consider two lines in a co-ordinate plane. Suppose they intersect at one point.

The two lines can be represented(??) as sets of ordered pairs formed by the x and y coordinates.

Since they intersect only at one point, the set-intersection of the sets that represent the two lines will be finite represented by the solution to the two simultaneous equations that represent the lines.

So, no, for two infinite sets A and B, it is not necessary that A n B has to be infinite.

Am I right?

Is there a better way to understand this problem? Thanks
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October 29th, 2015, 09:29 AM   #2
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Yes, you're right. The intersection of the even and odd numbers is empty hence finite as well.
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October 29th, 2015, 09:47 AM   #3
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Indeed, there are many good examples among the natural numbers. If you wish a non-zero intersection, consider A = the even numbers, B = the prime numbers.

Examples are even easier to find in the integers. A = the set of integers greater or equal to than the integer $a$, B= the set of integers less or equal to than the integer $b$ where $b \ge a$.
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Last edited by v8archie; October 29th, 2015 at 09:50 AM.
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