Set

 11.1 Set

 Definition A group of objects which have the common characteristics and classified in the same group.

Describe sets:

• Description
• Listing
• Set builder notation

 Example Describe the multiples of $$3$$ which are less than $$19$$ by using; (i) description (ii) listing (iii) set builder notation (i) Let the set be represented by $$R$$. Description: $$R$$ is the set of multiples of $$3$$ which are less than $$19$$. (ii) Listing: $$R=\{3, 6, 9, 12, 15, 18\}$$ (iii) Set builder notation: $$R=\{x:x\text{ is the multiple of 3 and }x\lt19\}$$

Empty set:

 Definition A set that contains no elements.

• An empty set can be represented with the symbol $$\phi$$ or { }.
• An empty set is also called a null set.
• The symbol $$\phi$$ is read as phi.

The element of a set:

 Definition Each object in a set.

• Each of the elements must satisfy the conditions of the set that is defined.
• Symbol $$\in$$ (epsilon) is used to represent ‘is an element of’ the set.
• Symbol $$\notin$$ is used to represent ‘is not an element of’ the set.

Determine the number of elements of a set:

• Number of elements in set $$P$$ can be represented by the notation $$n(P)$$.
• List all the elements in a set so that the number of elements in the set can be determined.

 Example Determine the number of elements in the following set. $$A=\{\text{colours of the traffic light\}}$$ Noted that \begin{aligned}A&=\{\text{colours of the traffic light\}} \\\\&=\{\text{red, yellow, green\}}. \end{aligned} Thus, $$n(A)=3$$.

Equality of sets:

 Definition Sets in which every element of the sets are the same.

• If every element in two or more sets are the same, then all the sets are equal.
• The order of elements in a set is not important.
• Symbol $$\neq$$ means ‘is not equal to’.

 Example Determine whether the following pair of sets is an equal set. \begin{aligned}S&=\{\text{letters in the word 'AMAN'}\} \\\\T&=\{\text{letters in the word 'MANA'}\} \end{aligned} We can see that \begin{aligned}S&=\{\text{A, M, A, N}\} \\\\T&=\{\text{M, A, N, A}\}. \end{aligned} Each element in set $$S$$ is equal to each element in set $$T$$. Thus, $$S=T$$.