Intersection of Sets

4.1

Intersection of Sets

Determine and Describe the Intersection of Sets Using Various Representations

Intersection of sets exist when there are more than one set.
The intersection of sets \(P\) and \(Q\) is written using the symbol \(\cap\).

Example

It is given that the universal set, \(\xi=\{x:x\text{ is an integer, }1\leq x\leq 10\}\), set \(P=\{x:x\text{ is an odd number}\}\), set \(Q=\{x:x\text{ is a prime number}\}\) and
set \(R=\{x:x\text{ is a multiple of 3}\}\).

List all the elements of the \(P\,\cap\,Q\), \(P\,\cap\,R\), \(Q\,\cap\,R\) and \(P\,\cap\,Q\, \cap R\).

	Answer
\(P\,\cap\,Q\)	\(P=\{1,3,5,7,9\}\) \(Q=\{2,3,5,7\}\) \(P\,\cap\,Q=\{3,5,7\}\)
\(P\,\cap\,R\)	\(P=\{1,3,5,7,9\}\) \(R=\{3,6,9\}\) \(P\,\cap\,R=\{3,9\}\)
\(Q\,\cap\,R\)	\(Q=\{2,3,5,7\}\) \(R=\{3,6,9\}\) \(Q\,\cap\,R=\{3\}\)
\(P\,\cap\,Q\, \cap R\)	\(P=\{1,3,5,7,9\}\) \(Q=\{2,3,5,7\}\) \(R=\{3,6,9\}\) \(P\,\cap\,Q\, \cap R=\{3\}\)

State the number of elements of the \(n(P\,\cap\,Q)\), \(n(P\,\cap\,R)\), \(n(Q\,\cap\,R)\) and \(n(P\,\cap\,Q\, \cap R)\).

	Answer
\(n(P\,\cap\,Q)\)	\(P\,\cap\,Q=\{3,5,7\}\) \(n(P\,\cap\,Q)=3\)
\(n(P\,\cap\,R)\)	\(P\,\cap\,R=\{3,9\}\) \(n(P\,\cap\,R)=2\)
\(n(Q\,\cap\,R)\)	\(Q\,\cap\,R=\{3\}\) \(n(Q\,\cap\,R)=1\)
\(n(P\,\cap\,Q\, \cap R)\)	\(P\,\cap\,Q\, \cap R=\{3\}\) \(n(P\,\cap\,Q\, \cap R)=1\)

The Intersections of Two or More Sets Using Venn Diagrams

Tips

\(B\subset A\): Set \(B\) is a subset of set \(A\) when all the elements of set \(B\) are found in set \(A\).
An empty set is a set that has no element and is represented by the symbol \(\phi\) or \(\{\}\).

Example

It is given that set \(A=\{\text{numbers on a dice}\}\), set \(B=\{\text{even numbers on a dice}\}\) and set \(C=\{7,8,9\}\).

List all the elements of the \(A\,\cap\,B\), \(B\,\cap\,C\) and \(A\,\cap\,C\).

Solution:

\(A=\{1,2,3,4,5,6\}\)
\(B=\{2,4,6\}\)
\(C=\{7,8,9\}\)

	Answer
\(A\,\cap\,B\)	\(A\,\cap\,B=\{2,4,6\}\)
\(B\,\cap\,C\)	\(B\,\cap\,C=\{\}\)
\(A\,\cap\,C\)	\(A\,\cap\,C=\phi\)

Draw a Venn diagram to represent sets \(A,B\) and \(C\), and shade the region that represents \(A\,\cap\,B\) and \(B\,\cap\,C\).

Solution:

\(A\,\cap\,B\)

\(B\,\cap\,C\)

All the elements of set \(B\) are in set \(A\).
\(A\,\cap\,B=B\).

Set \(B\) and set \(C\) do not have common elements.

All the elements of set \(B\) are in set \(A\).
\(A\,\cap\,B=B\).

The Complement of an Intersection of Sets

It is written using the symbol " \('\) ".
\((A\,\cap B)'\) is read as " the complement of the intersection of sets \(A\) and \(B\) ".
\((A\,\cap B)'\) refers to all the elements not in the intersection of sets \(A\) and \(B\).

Example

Given the universal set, \(\xi=\{x:x\text{ is an integer, }1\leq x\leq 8\}\), set \(A=\{1,2,3,4,5,6\}\), set \(B=\{2,4,6\}\) and set \(C=\{1,2,3,4\}\), list all the elements and state the number of elements of the \((A\,\cap\,B)'\), \((A\,\cap\,C)'\) and \((A\,\cap\,B\,\cap C)'\).

Solution:

\(\xi=\{1,2,3,4,5,6,7,8\}\)

\((A\,\cap\,B)'\)	\(A\,\cap\,B=\{2,4,6\}\) \((A\,\cap\,B)'=\{1,3,5,7,8\}\) \(n(A\,\cap\,B)'=5\)
\((A\,\cap\,C)'\)	\(A\,\cap\,C=\{1,2,3,4\}\) \((A\,\cap\,C)'=\{5,6,7,8\}\) \(n(A\,\cap\,C)'=4\)
\((A\,\cap\,B\,\cap C)'\)	\(A\,\cap\,B\,\cap C=\{2,4\}\) \((A\,\cap\,B\,\cap C)'=\{1,3,5,6,7,8\}\) \(n(A\,\cap\,B\,\cap C)'=6\)

The Complements of The Intersections of Two or More Sets On Venn Diagrams

Example

The co-curricular activities participated by three pupils are given in set \(P\), set \(Q\) and set \(R\) such that the universal set,
\(\xi=\{\text{Scouts, Mathematics, Hockey, Football, History, Badminton, Police Cadet\}}\).

\(P=\{\text{Scouts, Mathematics, Hockey\}}\)
\(Q=\{\text{Police Cadet, History, Badminton\}}\)
\(R=\{\text{Scouts, History, Fotball\}}\)

List all the elements of \((P\,\cap R)'\), \((R\,\cap Q)'\) and \((P \cap R \cap Q)'\) and draw a Venn diagram to represent sets \(P\), \(Q\) and \(R\), and shade the region that represents each of \((P\,\cap R)'\), \((R\,\cap Q)'\) and \((P \cap R \cap Q)'\).

Solution:

\(\xi=\{\text{Scouts, Mathematics, Hockey, Football, History, Badminton, Police Cadet\}.}\)

\((P\,\cap R)'\)
\((P\,\cap R)=\{\text{Scouts}\}\) \((P\,\cap R)'=\{\text{Mathematics, Hockey, Football, History, Badminton, Police Cadet\}}\)

\((R\,\cap Q)'\)
\((R\,\cap Q)=\{\text{History}\}\) \((R\,\cap Q)'=\{\text{Scouts, Mathematics, Hockey, Football, Badminton, Police Cadet\}}\)

\((P \cap R \cap Q)'\)
\((P \cap R \cap Q)=\{\,\}\) \((P \cap R \cap Q)'=\{\text{Scouts, Mathematics, Hockey, Football, History, Badminton, Police Cadet\}}\)

Solve Problem Involving The Intersections of Sets

Example

A total of \(140\) Form \(5\) pupils are given the opportunity to attend the intensive classes for History and Bahasa Melayu subjects. \(65\) pupils choose Bahasa Melayu, \(70\)pupils choose History while \(50\) pupils choose both Bahasa Melayu and History. Calculate the total number of pupils who attend the intensive classes and the total number of pupils who do not attend any intensive classes.

Solution:

\(\xi=\text{total number of puplis}=140\) \(n(B)=\text{puplis who attend Bahasa Melayu class}=65\) \(n(H)=\text{puplis who attend History class}=70\) \(n(B \cap H)=\text{pupils who attend both Bahasa Melayu and History classes}=50\)
Pupils who attend Bahasa Melayu Class only	\(65-50=15\)
Pupils who attend History Class only	\(70-50=20\)
Pupils who attend the intensive classes	\(15+50+20=85\)
Pupils who do not attend any intensive classes	\(140-85=55\)

Total number of pupils who attend the intensive classes = \(85\)

Total number of pupils who do not attend any intensive classes = \(55\)