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This is a problem from Discrete Mathematics and its Applications enter image description here

Here is my book's definition on countable enter image description here

and definition of having the same cardinality enter image description here

The only example that my book gave of uncountable set was the set of real numbers. I understand that because if you try listing out all of the members of the set, you would keep going on and on - 1, 1.01, 1.001, etc...... But the intersection of the set of real numbers and itself is the set of real numbers is uncountable as well... Is there another uncountable set that you could use to prove this?

Lehs
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committedandroider
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3 Answers3

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Here's another idea,

$$(-\infty,0]\cap[0,\infty)=\{0\}$$

What can you say about the cardinality of $(-\infty,0]$ and $[0,\infty)$?

Tim Raczkowski
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  • The cardinality of (−∞,0] and [0,∞) would be 1 because 0 is the only element for which they intersect. so the intersection set would just be {0} and the size or cardinality of that set is just 1. – committedandroider Jan 27 '15 at 20:46
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Here's one idea: suppose $C$ and $D$ are disjoint uncountable sets, and $E$ is finite. Consider $A=C \cup E$ and $B=D \cup E$. Then $A \cap B=E$ is finite. Can you come up with two disjoint uncountable sets to set this up?

Ian
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  • Why bother with $E$? $\varnothing$ is finite. – Brian M. Scott Jan 20 '15 at 19:36
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    @BrianM.Scott Fair point. This is a little bit nice because it shows the cardinality could be any number of things that are smaller than the cardinality of $C$ and $D$ themselves. – Ian Jan 20 '15 at 19:37
  • And disjoint means their intersection is the empty set right? – committedandroider Jan 20 '15 at 21:01
  • @committedandroider Correct. – Ian Jan 20 '15 at 21:02
  • Is the shortcut I used to determine if its countable or not a good one? Try to start to listing them out - 1,2,3,4 vs 1.001, 1.0001. I wasn't sure if this shortcut is a mathematically correct one to use – committedandroider Jan 20 '15 at 21:03
  • @committedandroider Actually no, because there could be a complete list and an incomplete list, and if your example is the incomplete list then you may incorrectly conclude that there is no complete list. Consider for example the failure to list all the natural numbers with the list 2,4,6,... You should look up Cantor's diagonal argument. – Ian Jan 20 '15 at 21:07
  • @Ian Is there a shortcut to that argument you see? Now when I look at something like [1,4], I can just tell that is uncountable – committedandroider Jan 27 '15 at 20:47
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Can you give an example of an uncountable set $B$ that is disjoint from the uncountable set $A= [0,1)$ ??

HexedAgain
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  • Two sets are disjoint if their intersection is the empty set. So in this case, I come up with B = (100,101). The intersection of [0,1) and (100,101) would be the empty set {} which has a cardinality or size of 0. – committedandroider Jan 27 '15 at 20:50