I'm hoping someone can explain how I go about drawing up venn diagrams in my discrete structures class. The book this class uses doesn't explain much at all, and gives only two examples..

My homework asks me to:

Use Venn diagrams to determine whether each of the following is true or false:

  1. $(A ∪ B) ∩ C = A ∪ (B ∩ C)$
  2. $A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)$

I'm NOT looking for the answers to these, I just need a better understanding on how I can draw these out so I can answer them.

Would these look correct?

  1. $A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)$ = true from what my drawings are showing below

(A ∩ B) ∪ (A ∩ C)

A ∩ (B ∪ C)

  • 1
    $\begingroup$ Why do you think the drawings indicate that $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ is false? $\endgroup$
    – Steve Kass
    Jun 18, 2016 at 22:12

2 Answers 2


Here is how I would approach the problem. For each of these equations, start by drawing two Venn diagrams with three sets each. Label the sets A, B, and C. I'm thinking of something that looks like this.

$\hskip2in$ venn_diagram

In the first Venn diagram, shade the region corresponding to the set on the left hand side of the equation. Similarly, in the second Venn diagram, shade the region corresponding to the set on the right hand side of the equation. If the shadings match up, then the equation is correct. This exercise provides a nice visual intuition for why these statements may or may not be true.

  • $\begingroup$ I drew out each side for question two, if you get a chance, I could use someone to look over it! I believe question two is false? but I might be wrong $\endgroup$
    – Rickybobby
    Jun 18, 2016 at 21:28
  • $\begingroup$ Your Venn diagrams aren't quite right. Your shading for $(A \cap B) \cup (A \cap C)$ is correct. However, your shading for $A \cap (B \cup C)$ is incorrect. Instead, you have shaded $B \cup C$. Don't forget to take the intersection with $A$. Make this change and then see what you get. $\endgroup$ Jun 18, 2016 at 21:52
  • $\begingroup$ Updated the second one! Looks like both are then equal I believe $\endgroup$
    – Rickybobby
    Jun 18, 2016 at 21:56
  • $\begingroup$ Great work! You have now shown that set intersection distributes over set union. In the same way that multiplication distributes over addition -- that is, $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ -- set intersection distributes over set union. Feel free to accept the answer if you found it helpful! $\endgroup$ Jun 18, 2016 at 22:04

In both of the problems you've given, you only have three sets: $A, B,$ and $C$.

For each problem, start by drawing out a labeled, three-set Venn diagram for each side of the equation and lightly shade over portions of the diagram that would correspond to the given set operations.

You may consider drawing more than one diagram for each side of the equation, to better show the order of operations that the parentheses requires you to take. For example, given problem 1, first draw one three-set diagram shading the result of $(A \cup B)$ and then draw a second diagram for the final shading of $(A \cup B) \cap C$, and repeat for the right side of the equation.

Here are two diagrams for the union and intersection of 2 sets, posted here, which may help you understand how you need to shade the diagrams. You can find many more examples of shaded three-set Venn diagrams online.


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