11.1  Set

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

Describe sets:
  • Description
  • Listing
  • Set builder notation

Describe the multiples of \(3\) which are less than \(19\) by using;

(i) description

(ii) listing

(iii) set builder notation


Let the set be represented by \(R\).

Description: \(R\) is the set of multiples of \(3\) which are less than \(19\).


Listing: \(R=\{3, 6, 9, 12, 15, 18\}\)


Set builder notation:

\(R=\{x:x\text{ is the multiple of 3 and }x\lt19\}\)

Empty set:

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:

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.

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:

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’.

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, \(S=T\).