The relative size of Cardinalities of Sets

Question

Let A and B be sets in a finite universal set U. List the following in order of increasing size.

In this question, I don't understand why $|A\cup B| \geq |A-B|$.

In my logic,

$A-B=A\cap B^c$

$B^c=U-B=U\cap B^c$

Therefore, I think $|U\cap B^c| \geq |A\cup B|$

I have no idea why my logic is wrong. Please provide me an explanation.

Thank you in advance.

Neither is correct. It depends on what $U$, $A$ and $B$ are. For instance, if $U=\{1,2,\ldots, 10\}$ and $A = \{1,\ldots,5\}$ and $B = \{6,\ldots,10\}$, then $|A\cup B| = 10 > |U \cap B^c| = 5$. Whereas if $A = \{1\}$ and $B = \{2\}$, then $|A \cup B| = 2 < |U \cap B^c| = 9$ — User, Commented Sep 4, 2023 at 13:46

Eric · Accepted Answer · 2023-09-05 13:16:35Z

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I found my mistakes.

$A-B=A\cap B^c=A\cap U\cap B^c, $ not $A\cup U\cap B^c$.

Therefore, $|A\cup B|\geq|A-B|$ is correct.

answered Sep 5, 2023 at 13:16

Eric

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$\begingroup$ That is the reason your original analysis was wrong. The reason $|A\cup B| \ge |A - B|$ is correct is Gerry Myerson's comment. $\endgroup$
– Paul Sinclair
Commented Sep 5, 2023 at 15:43

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