5.2 A Fundamental Principle of Counting

1. Number of Elements in A2. Inclusion-Exclusion Principle3. Venn Diagram4. De Morgan's Laws

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Number of Elements in A

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If S is any set, we will denote the number of elements in S by n(S). For example, if S = {1,7,11}, then n(S) = 3;if S = , then n(S) = 0.

Inclusion-Exclusion Principle

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Inclusion-Exclusion PrincipleLet S and T be sets. Then

( ) ( ) ( ) ( ).n S T n S n T n S T

Example Inclusion-Exclusion Principle

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In the year 2000, Executive magazine surveyed the presidents of the 500 largest corporations in the US. Of these 500 people, 310 had degrees (of any sort) in business, 238 had undergraduate degrees in business, and 184 had graduate degrees in business. How many presidents had both undergraduate and graduate degrees in business?

Example Inclusion-Exclusion Principle (2)

Let S = {presidents with an undergraduate degree in business} and T = {presidents with a graduate degree in business}.S T = {presidents with degrees (of any sort) in business}S T = {presidents with both undergraduate and graduate degrees in business}

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Example Inclusion-Exclusion Principle (3)

n(S) = 238n(T) = 184

n(S T) = 310n(S T) = n(S) + n(T) n(S T)

310 = 238 + 184 n(S T)n(S T) = 238 + 184 310 = 112

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Venn DiagramA Venn diagram is a drawing that represents sets geometrically. To construct a Venn diagram, draw a rectangle and view its points as the elements of U. Then draw a circle inside the rectangle for each set. The circles should overlap. View the points inside the circles as elements of each set.

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Example Venn DiagramDraw a Venn diagram with three sets, R, S and T. Shade the area that represents R S T, R S T and S R' T '.

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Example Venn Diagram (2)

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De Morgan's Laws

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De Morgan's Laws Let S and T be sets. Then(S T)' = S' T',(S T)' = S' T'.

Example De Morgan's Laws

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Verify (S T)' = S ‘ T ' using Venn diagrams.

Example De Morgan's Laws (2)

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Example De Morgan's Laws (3)

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Summary Section 5.2 - Part 1

The inclusion-exclusion principle says that the number of elements in the union of two sets is the sum of the number of elements in each set minus the number of elements in their intersection.

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Summary Section 5.2 - Part 2

A Venn diagram consists of a rectangle containing overlapping circles and is used to depict relationships among sets. The rectangle represents the universal set and the circles represent subsets of the universal set. De Morgan's laws state that the complement of the union (intersection) of two sets is the intersection (union) of their complements.

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