Given that there are 100 elements in $A_1$ , 1000 elements in $A_2$ , and 10,000 elements in $A_3$, find the number of elements in $A_1 \cup A_2 \cup A_3$ in each of the following cases.

(a) $A_1 ⊆ A_2$ and $A_2 ⊆ A_3$ ;

(b) the sets are pairwise disjoint;

(c) there are 2 elements common to each pair of sets and 1 element in all three sets.

Draw, and use, relevant Venn diagrams in each case.

This my working out:

a) $A_1 ⊆A_2 \implies A_1 \cup A_2 = A_2$ and $A_2 ⊆A_3 \implies A_2 \cup A_3 = A_3$

Therefore: $A_1 \cup A_2 \cup A_3 \overset{\tiny\text{(associative property)}}= (A_1 \cup A_2) \cup A_3 \overset{\tiny\text{(see above)}}= A_2 \cup A_3 \overset{\tiny\text{(see again above)}}= A_3$, so the union is the set A3 and therefore contains 10000 elements.

b) the sets are pairwise disjoint By the inclusion-exclusion principle, where N(X) = the number of elements in the finite set X

$\begin{align}N(A_1 \cup A_2 \cup A_3) ~=~& {N(A_1) + N(A_2) + N(A_3) \\- N(A_1 \cap A_2) - N(A_1 \cap A_3) - N(A_2 \cap A_3) \\+ N(A_1 \cap A_2 \cap A_3)}\\ ~\overset{\tiny\text{(Since they're pairwise disjoint, any intersection is the empty set)}}{\qquad\qquad\qquad\qquad\quad=}~& N(A_1) + N(A_2) + N(A_3) - 0 - 0 - 0 + 0 \\ ~=~& 100 + 1000 + 10000 \\~=~& 11100\text{ elements}\end{align}$

c) Using the inclusion-exclusion principle again:

$\begin{align}N(A_1 \cup A_2 \cup A_3) ~=~& {N(A_1) + N(A_2) + N(A_3) \\- N(A_1 \cap A_2) - N(A_1 \cap A_3) - N(A_2 \cap A_3) \\+ N(A_1 \cap A_2 \cap A_3)}\\ =~& 100 + 1000 + 10000 - 2 - 2 - 2 + 1 \\ =~& 11100 - 5 \\=~& 11095\end{align}$

However I am not sure how to convey this through Venn Diagrams. Also, can someone tell me if this correct. Thanks :)

  • $\begingroup$ Fo c), use inclusion exclusion principle ! (but in b) is not necessary, since If $A_i$ are disjoint, then $|A_1\cup...\cup A_n|=|A_1|+...+|A_n|$.) $\endgroup$ – Surb Oct 9 '16 at 7:10
  • $\begingroup$ so there is no need to use inclusion exclusion principle for b)? but it is needed for c? $\endgroup$ – Math.10 Oct 9 '16 at 7:14
  • $\begingroup$ As stated in the problem, draw a Venn diagram. You know how a Venn diagram of three sets looks, right? Then then information in the problem tells you that certain areas in the Venn diagram are empty or have a specific number of elements in them, and then you can puzzle out what the number of elements in the other areas must be one by one, such as to make the total sizes of each set come out right. $\endgroup$ – Henning Makholm Oct 9 '16 at 7:14
  • $\begingroup$ No for b) but yes for c) $\endgroup$ – Surb Oct 9 '16 at 7:15
  • $\begingroup$ There is no need to use symbolic inclusion-exclusion at all if you do as you're told and use a Venn diagram. $\endgroup$ – Henning Makholm Oct 9 '16 at 7:16

Your answers are correct if you were asked to obtain them through symbolic logic. However you were asked to obtain them via Venn Diagrams.

a) The Venn Diagram for $A_1\subseteq A_2\subseteq A_3$ is simply a set of matryoshka-doll style ovals (or elipses); with the smaller sets entirely enclosed inside the larger. $$\require{enclose}\color{red}{\enclose{circle}{\color{green}{\enclose{circle}{\color{blue}{\enclose{circle}{A_1}}A_2}}A_3}}$$

b) For Pairwise Disjoint, you already know this means that the sets have no intersections. Draw three non-intersecting ovals.

c) For this you have three sets intersecting with given overlap sizes. You have enough information to determine the size of all seven disjoint areas of the union.


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