Venn-Euler Diagrams & Laws On Operations On Set - SS1 Mathematics Lesson Note

Two mathematicians John Venn and Leonard Euler popularized a simple yet instructive system for understanding set operations called Venn-Euler Diagrams.

Figure 1

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Figure 2

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Figure 3

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Figure 4

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Figure 5

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Figure 6

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In figure 1, \(A \subset U,\ B \subset U,\ C \subset U\ and\ C \subset B\). Also, \(B \supset C\).

In figure 2, \(A\ and\ B\) are disjoint sets meaning \(A \cap B = \{\}\).

In figure 3, the regions \(\mathbf{I\ and\ II}\)are part of set \(\mathbf{A}\ \)while regions \(\mathbf{II\ and\ III}\) form part of set \(\mathbf{B}\). Thus, regions \(\mathbf{I,\ II\ and\ III}\) together denote \(\mathbf{A \cup B}\).

In figure 4, \(C = \ A \cap B\).

In figure 5, region \(\mathbf{I\ }\)only denotes \(\mathbf{A - B.}\) Region \(\mathbf{III\ }\)only denotes \(\mathbf{B - A}\).

In figure 6, \(A\ and\ A'\) are shown.

Laws On Operations On Set

1. \(\left( A^{'} \right)^{'} = A\)

\(\varnothing^{'} = U\)

\(A - A = \varnothing;\ \ A - \varnothing = A\)

\(A \cup \varnothing = A\)

\(A \cup A = A\)

\(A \cup A^{'} = U\)

\(A \cup U = U\)

\(U^{'} = \varnothing\)

\(A \cap \varnothing = \varnothing\)

\(A \cap A = A\)

\(A \cap A^{'} = \varnothing\)

\((A \cup B)^{'} = A' \cap B'\)

\((A \cap B)^{'} = A' \cup B'\)

\(A - B = A \cap B^{'} = B^{'} - A\)

\(A - B = B - A;if\ and\ only\ if\ A = B\)

\(A - B = \varnothing;if\ and\ only\ if\ A \subset B\)

\(A - B = A;if\ and\ only\ if\ A \cap B = \varnothing\)

\(A \cup B = B \cup A\)

Application of Venn-Euler Diagrams

Example 1 At the matriculation ceremony of a university 800 students showed up. During a football match between two departments, 600 students turned up. If there are 1075 students enrolled in the university, how many attended both functions?

Solution

The universal set \(U\) is the number of students enrolled in the university

\[n(U) = 1075\]

\(n(M) = 800\), number of students at the matriculation ceremony.

\(n(F) = 600\), numbers of students at the football match.

\(n(M \cap F) = x\), number that attended both.

\[n(U) = \ n(M) + n(F) - n(M \cap F)\]

\[1075 = 800 + 600 - x\]

\[1075 = 1400 - x\]

\[1075 - 1400 = - x\]

\[- 325 = - x\]

\[x = 325\]