Set notation and Venn diagrams 1
What is set notation?
Set notation is known as a collection of elements specified to determine whether a given element is contained in a set or not. This means that $$a$$ is an element of $$X$$ or $$a$$ belongs to set $$X$$. This can be written as $$ a\in X$$. The notation is $$ a\notin X$$, which means that $$a$$ does not belong to $$X$$.
Set notations are the symbolic representations of elements that talk about the properties of sets.
E1.2: Use language, notation and Venn diagrams to describe sets and represent relationships between sets
Set notation defines the elements of a set and illustrates the relationship and operations among sets. Now, let us discuss the symbols and their meanings. Let $$\in$$ mean ‘is an element of set’, $$ \notin$$ mean ‘is not an element of set’, $$ \left \{ \right \}$$ denote a set $$ \mid$$ and $$:$$ symbols mean ‘such that’ or ‘for which’.
Now, discuss how to read and write the set notation.
A set and its elements
A set is denoted by capital letters and the elements are represented by lowercase letters. The elements are separated by using a comma.
For example, let a set be $$ X=\left \{ a, e, i, o, u \right \}$$. It can be read as a set $$X$$ that contains the English vowels.
Set membership
Set membership states if the elements are contained in a set or not. The symbol $$ \in$$ is used for representing that an element is contained in a set and the symbol $$ \notin$$ is used for representing that an element is not contained in a set.
For example: Let a set be $$ Y=\left \{2,4,6,8 \right \}$$. So, we can write that $$ 2\in Y$$ means $$2$$ is an element of $$Y$$ and $$1\notinY$$ means $$1$$ is not an element of $$Y$$.
Specify member of a set
The set can be described in the set builder form as shown below:
$$P=\left \{ x\mid x\in \mathbb{R} \;\text{and}\; x\leq 3 \right \}$$ or $$ P=\left \{ x: x\in \mathbb{R} \;\text{and}\; x\leq 3 \right \}$$.
It can be read that the set $$P$$ contains all the values which belong to real number and also are less than equal to $$3$$.
Worked examples of sets and Venn diagrams
Example 1: $$X=\left \{ 1,3,5,7 \right \}$$ and $$Y=\left \{ 21,47 \right \}$$. Find $$X\cap Y$$ and $$n\left (X\cap Y \right )$$.
Step 1: Write the given values.
$$ X=\left \{ 1,3,5,7 \right \}$$ and $$ Y=\left \{ 21,47 \right \}$$.
Step 2: Find $$X\cap Y$$.
Since there are no common elements in both sets, the result is a null set or empty set.
So, $$ X\cap Y=\left \{ \right \}$$.
Step 3: Find the value of $$n\left (X\cap Y \right )$$.
Since it is a null set, the value of $$n\left (X\cap Y \right )$$ is $$0$$.
Example 2: $$A=\left \{ 1, 2, 3, 4 \right \}$$ and $$B=\left \{ 4,5,6 \right \}$$. Find $$A\cup B$$ and also $$n\left ( A \right )$$, $$n\left ( B \right )$$ and $$n\left ( A\cup B \right )$$.
Step 1: Write the given values.
$$A=\left \{ 1, 2, 3, 4 \right \}$$ and $$B=\left \{ 4,5,6 \right \}$$.
Step 2: Find $$A\cup B$$.
$$A\cup B=\left \{ 1, 2, 3, 4, 5, 6 \right \}$$.
Step 3: Find $$n\left ( A \right )$$.
The number of elements in $$A$$ is $$4$$. Therefore, $$n\left ( A \right)=4$$.
Step 4: Find $$n\left ( B \right )$$.
The number of elements of $$B$$ is $$3$$. Therefore, $$n\left ( B \right )=3$$.
Step 5: Find $$n\left ( A\cup B \right )$$.
$$A\cup B=\left \{ 1, 2, 3, 4, 5, 6 \right \}$$.
The number of elements in $$A\cup B$$ is $$6$$. Therefore, $$n\left ( A\cup B \right )=6$$.