Combined Operation of Set

Combined Operations on Sets

4.3

Combined Operations on Sets

Involve both intersection of sets \(( \cap )\) and union of sets \(( \cup )\) at the same time.

Example

The table below shows the hobbies of a group of pupils.

Arif likes singing.	Iris likes drawing, singing and dancing.
Zarif likes drawing.	Alan likes singing and dancing.
Lily likes dancing.	May likes dancing and drawing.
Emy likes dancing and singing.	Jay likes singing and drawing.
Getha likes drawing.	Nani likes dancing.

\(P=\{\text{pupils who like singing\}}\)
\(Q=\{\text{pupils who like dancing\}}\)
\(R=\{\text{pupils who like drawing\}}\)

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

Solution:

\(P=\{\text{Arif, Emy, Iris, Alan, Jay\}}\)
\(Q=\{\text{Lily, Emy, Iris, Alan, May, Nani\}}\)
\(R=\{\text{Zarif, Getha, Iris, May, Jay\}}\)

\((P\,\cup \,Q)\,\cap\,R\)	\(\qquad\;(P\,\cup \,Q)=\{\text{Arif, Emy, Iris, Alan, Jay, Lily, May, Nani}\}\\\quad \quad \quad \quad \quad \,\,R=\{\text{Zarif, Getha, Iris, May, Jay}\}\\(P\,\cup \,Q)\,\cap\,R=\{\text{Jay, Iris, May}\}\)
\(Q\cup(P\,\cap \,R)\)	\(\qquad \qquad \quad Q=\{\text{Lily, Emy, Iris, Alan, May, Nani}\}\\\quad\quad\;(P\cap R)=\{\text{Iris, Jay}\}\\Q\cup(P\,\cap \,R)=\{\text{Lily, Emy, Iris, Alan, May, Nani, Jay}\}\)

The Complement of Combined Operations on Sets

Involve the complement of either intersection of sets \((\cap)\) or union of sets \((\cup)\) or both intersections.

Example

It is given that \(\xi=\{x:x\text{ is an integer, }30\le x\le40\}\), set \(A=\{x:x\text{ is a multiple of 3}\}\), set \(B=\{x:x\text{ is a number such that the sum of its two digits is odd}\}\) and set \(C=\{30,32,35,39,40\}\).

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

Solution:

\(\;\xi=\{30,31,32,33,34,35,36,37,38,39,40\}\)
\(A=\{30,33,36,39\}\)
\(B=\{30,32,34,36,38\}\)
\(C=\{30,32,35,39,40\}\)

\((A\,\cup \,B)'\,\cap\,C\)	\(\qquad\;(A\,\cup \,B)'=\{31,35,37,40\}\\\quad \quad \quad \quad \quad \;\;C=\{30,32,35,39,40\}\\(A\,\cup \,B)'\,\cap\,C=\{35,40\}\)
\(A'\cap(B\,\cup \,C)\)	\(\qquad \qquad \;\;\;A'=\{31,32,34,35,37,38,40\}\\\quad\quad(B\,\cup \,C)=\{30,32,34,35,36,38,39,40\}\\A'\cap(B\,\cup \,C)=\{32,34,35,38,40\}\)
\((A\cap C)'\cup(B\,\cap \,C)\)	\(\qquad\qquad\quad(A\cap C)'=\{31,32,33,34,35,36,37,38,40\}\\\quad \quad\quad\quad\;\;\;(B\,\cap \,C)=\{30,32\}\\(A\cap C)'\cup(B\,\cap \,C)=\{30,31,32,33,34,35,36,37,38,40\}\)

Solve Problem Involving Combined Operations on Sets

The Residents's Association of Happy Garden organises various sports competitions to instil health awareness among residents. A total of \(35\) participants join the football competition, \(24\) participants join the table tennis competition and \(13\) participants join the badminton competition. There are \(4\) participants who join both the football and table tennis competitions, \(8\) participants who join both the table tennis and badminton competitions, and \(2\) participants join all three competitions. There is no participant joining the badminton and football competitions only. Calculate the total number of participants who join one competition only.

Solution:

	Total
Number of Participants	\(?\)
Football	\(35\)
Table Tennis	\(24\)
Badminton	\(13\)
Football & Table Tennis	\(4\)
Badminton & Football	\(0\)
Table Tennis & Badminton	\(8\)
Football, Table Tennis & Badminton	\(2\)
Number of Participants Who Join One Competition Only	\(?\)

\(\;\xi=\{\text{all participants}\}\)
\(A=\{\text{participants who join the football competition}\}\)
\(B=\{\text{participants who join table tennis competition}\}\)
\(C=\{\text{participants who join the badminton competition}\}\)

Football only:
\(35-4=31\)

Badminton only:
\(13-6-2=5\)

Table tennis only:
\(24-6-2-2=14\)

Total number of participants who join one competition only:
\(31+14+5=50\)

Combined Operations on Sets

4.3

Combined Operations on Sets

Involve both intersection of sets \(( \cap )\) and union of sets \(( \cup )\) at the same time.

Example

The table below shows the hobbies of a group of pupils.

Arif likes singing.	Iris likes drawing, singing and dancing.
Zarif likes drawing.	Alan likes singing and dancing.
Lily likes dancing.	May likes dancing and drawing.
Emy likes dancing and singing.	Jay likes singing and drawing.
Getha likes drawing.	Nani likes dancing.

\(P=\{\text{pupils who like singing\}}\)
\(Q=\{\text{pupils who like dancing\}}\)
\(R=\{\text{pupils who like drawing\}}\)

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

Solution:

\(P=\{\text{Arif, Emy, Iris, Alan, Jay\}}\)
\(Q=\{\text{Lily, Emy, Iris, Alan, May, Nani\}}\)
\(R=\{\text{Zarif, Getha, Iris, May, Jay\}}\)

\((P\,\cup \,Q)\,\cap\,R\)	\(\qquad\;(P\,\cup \,Q)=\{\text{Arif, Emy, Iris, Alan, Jay, Lily, May, Nani}\}\\\quad \quad \quad \quad \quad \,\,R=\{\text{Zarif, Getha, Iris, May, Jay}\}\\(P\,\cup \,Q)\,\cap\,R=\{\text{Jay, Iris, May}\}\)
\(Q\cup(P\,\cap \,R)\)	\(\qquad \qquad \quad Q=\{\text{Lily, Emy, Iris, Alan, May, Nani}\}\\\quad\quad\;(P\cap R)=\{\text{Iris, Jay}\}\\Q\cup(P\,\cap \,R)=\{\text{Lily, Emy, Iris, Alan, May, Nani, Jay}\}\)

The Complement of Combined Operations on Sets

Involve the complement of either intersection of sets \((\cap)\) or union of sets \((\cup)\) or both intersections.

Example

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

Solution:

\(\;\xi=\{30,31,32,33,34,35,36,37,38,39,40\}\)
\(A=\{30,33,36,39\}\)
\(B=\{30,32,34,36,38\}\)
\(C=\{30,32,35,39,40\}\)

\((A\,\cup \,B)'\,\cap\,C\)	\(\qquad\;(A\,\cup \,B)'=\{31,35,37,40\}\\\quad \quad \quad \quad \quad \;\;C=\{30,32,35,39,40\}\\(A\,\cup \,B)'\,\cap\,C=\{35,40\}\)
\(A'\cap(B\,\cup \,C)\)	\(\qquad \qquad \;\;\;A'=\{31,32,34,35,37,38,40\}\\\quad\quad(B\,\cup \,C)=\{30,32,34,35,36,38,39,40\}\\A'\cap(B\,\cup \,C)=\{32,34,35,38,40\}\)
\((A\cap C)'\cup(B\,\cap \,C)\)	\(\qquad\qquad\quad(A\cap C)'=\{31,32,33,34,35,36,37,38,40\}\\\quad \quad\quad\quad\;\;\;(B\,\cap \,C)=\{30,32\}\\(A\cap C)'\cup(B\,\cap \,C)=\{30,31,32,33,34,35,36,37,38,40\}\)

Solve Problem Involving Combined Operations on Sets

Solution:

	Total
Number of Participants	\(?\)
Football	\(35\)
Table Tennis	\(24\)
Badminton	\(13\)
Football & Table Tennis	\(4\)
Badminton & Football	\(0\)
Table Tennis & Badminton	\(8\)
Football, Table Tennis & Badminton	\(2\)
Number of Participants Who Join One Competition Only	\(?\)

Football only:
\(35-4=31\)

Badminton only:
\(13-6-2=5\)

Table tennis only:
\(24-6-2-2=14\)

Total number of participants who join one competition only:
\(31+14+5=50\)

Combined Operations on Sets

Combined Operations on Sets

Chapter : Set Operation

Topic : Combined Operation of Set

Related notes