4.3: Venn Diagrams
- Page ID
- 30260
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- Use a Venn diagram to illustrate the relationship between 2 and 3 sets
- Create an expression relating 2 and 3 sets from a Venn diagram
To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called Venn Diagrams.
A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets.
Basic Venn diagrams can illustrate the interaction of two or three sets.
Create Venn diagrams to illustrate \(A \cup B, A \cap B,\) and \(A' \cap B\).
Solution
\(A \cup B\) contains all elements in either set.
\(A \cap B\) contains only those elements in both sets - in the overlap of the circles.
\(A' \cap B\) contains the elements of \(B\) that are not in \(A\).
Use a Venn diagram to illustrate \((H \cap F)' \cap W\).
Solution
We'll start by identifying everything in the set \(\mathrm{H} \cap F\).
Now, \((H \cap F)' \cap W\) will contain everything not in the set identified above that is also in set \(W\).
Create an expression to represent the outlined part of the Venn diagram shown.
Solution
The elements in the outlined set are in sets \(\mathrm{H}\) and \(F\), but are not in set \(W\). So we could represent this set as \(H \cap F \cap W' \).
Create an expression to represent the outlined portion of the Venn diagram shown
- Answer
-
\( \left( A \cup B \right) \cap C'\)