Venn diagrams and counting

Japanese school year: Math I

What you learn

You will learn how to visualize overlapping sets using Venn diagrams and calculate element counts with the inclusion-exclusion principle. This technique is widely used in survey data analysis and combinatorial counting. Prior familiarity with basic set operations like unions and intersections will be very beneficial.

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Key points

Use this formula to find the total count of items in $A$ or $B$ . Add the individual counts $n(A)$ and $n(B)$ , then subtract the overlapping intersection $n(A \cap B)$ to avoid double counting.

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Use this formula to find the number of items outside set $A$ . Subtract the count in $A$ , $n(A)$ , from the total universal count $n(U)$ .

n(\bar{A}) = n(U) - n(A)

Use this formula to find the total count of items in $A$ , $B$ , or $C$ . Add the three individual sets, subtract all two-set overlaps, and add back the center intersection shared by all three.

n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)

Choose a set to practice.

Practice 1Coming soon