I would like to know the best way to answer this question!

Assume that $| A ∩ B |= 13$, $| A |= 17$ and $| B |= 19$. Determine the value of $| A ∪ B |$.



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  • 3
    $\begingroup$ Start by drawing a Venn diagram. $\endgroup$ – Randall Apr 26 at 2:32

This is best done through the inclusion-exclusion principle. Venn diagrams are helpful as a visual. Formulaically, though, inclusion-exclusion gives us

$$|A \cup B| = |A| + |B| - |A \cap B|$$

Substitute in the appropriate values and you should be good to go.

  • $\begingroup$ Thank you so, 17+19 = 36 - 13 = 23? So the answer must be 23! $\endgroup$ – AGRHQ Apr 26 at 2:36
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    $\begingroup$ Well, to nitpick, you should write it as $17+19=36$ followed by $36-13=23$ (the way you wrote it suggests $17+19=23$, though I get what you meant by it). But yes, $23$ is correct. $\endgroup$ – Eevee Trainer Apr 26 at 2:37
  • $\begingroup$ Thank youuuuuuu $\endgroup$ – AGRHQ Apr 26 at 2:38

$A$ contains $17$ things. $B$ contains $19$ things. Some of those things are in both sets. $13$, in fact. So, $13$ of the things are counted twice if we add $|A|+|B|$. How can we correct the overestimate of $17+19=36$? Does this get you started?

  • $\begingroup$ Thank you I now understand 17+19=36, 36-13=23! $\endgroup$ – AGRHQ Apr 26 at 2:37
  • $\begingroup$ Eevee's answer is the formula, and mine is how it works. Both good to know. And you are welcome. $\endgroup$ – The Count Apr 26 at 2:38

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