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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

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 \(A' = \{x|x\in U \,and\,x\notin a\} \)

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 *B *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 *A *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

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}

**Solution**:

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.

- Last Updated: Aug 31, 2020 10:52 AM
- URL: https://libraryguides.centennialcollege.ca/c.php?g=717286
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