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Set Theory

Venn Diagrams

A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plan, and set as regions inside closed curves. 


The rectangle represents the universal set (all elements), U, the portion bounded by the circle represents all elements in set A. The complement of set AA', contains all elements that are NOT in set A, but are contained in U

For example, if set A represents all natural numbers less than 10 or \(\{x \in \mathbb{N}| x < 10\} \), and the universal set U, contains all the natural numbers. Then the Venn Diagram will look like the following.


The complement of set A in builder notation is represented by \(A' = \{x|x\in U \,and\,x\notin a\} \)

Subsets of a Set

Set A is a subset of set B if every element of A is also an element of B. In symbols this is written \(A \subset B\) or \(A \subseteq B\) 


For example, if you are representing all the countries in the world, and set A represents Finland and Greece, and set represents all countries in Europe. Then \(A \subseteq B\) and the Venn Diagram can be represented as:


If set C represents all the countries in Asia, then set is not a subset of set C, represented by the notation \(A \nsubseteq C\)

Set A is a proper subset of set B if set A is a subset of B, but not equal to B. A proper subset can be expressed \(A \subset B\) or \(A \subseteq B\), but a subset that is not a proper subset is written (A \subseteq B\).

For example, set {a, b, c} is a proper subset of {a, b, c, d}. Thus, it can be expressed as \(\{a, b, c\} \subset \{a, b, c, d\}\) or \(\{a, b, c\} \subseteq \{a, b, c, d\}\)

Set {4, 7, 10} is not a proper subset of {4, 7, 10}. This can be expressed only with \(\{4, 7, 10\} \subseteq \{4, 7, 10\}\)

\(\therefore \) the equal set is one of the subsets, but the equal set is not a proper subset

Number of Subsets

The number of subsets of a set with n elements is \(2^n\).

The number of proper subsets of a set with n elements is \(2^n - 1\). Where the \(-1\) represents one equal set being subtracted because it is not a proper subset.

Something to think about: Is the empty set \(\varnothing \) a subset?

Example: Find the number of subsets and the number of proper subsets of the set {M, A, T, H}


There are 4 elements, so the number of subsets \(2^4 = 16\) and the number of proper subsets \(2^4 - 1 = 15\).

Can you write all the subsets out? This will help you answer the question above of whether \( \varnothing \) a subset.

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