Union of Sets

4.2 Union of Sets
 
Union of Sets
  • The union of sets \(P\) and \(Q\) is written using the symbol \(\cup\).
  • \(P \cup Q\) represents all the elements in set \(P\) or set \(Q\) or in both sets \(P\) and \(Q\).
 
The Union of Two or More Sets Using The Venn Diagram
Example


It is given that set \(P=\{\text{factors of 24}\}\), set \(Q=\{\text{multiples of 3 which are less than 20}\}\) and set \(R=\{\text{multiples of 4 which are less than 20}\}\).

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

\(P \cup Q\) \(P=\{1,2,3,4,6,8,12,24\}\)
\(Q=\{3,6,9,12,15,18\}\)
\(P\,\cup Q=\{1,2,3,4,6,8,9,12,15,18,24\}\)
\(P \cup R\) \(P=\{1,2,3,4,6,8,12,24\}\)
\(R=\{4,8,12,16\}\)
\(P\,\cup R=\{1,2,3,4,6,8,12,16,24\}\)
\(Q \cup R\) \(Q=\{3,6,9,12,15,18\}\)
\(R=\{4,8,12,16\}\)
\(Q\,\cup R=\{3,4,6,8,9,12,15,16,18\}\)
\(P\cup Q \cup R\) \(P=\{1,2,3,4,6,8,12,24\}\)
\(Q=\{3,6,9,12,15,18\}\)
\(R=\{4,8,12,16\}\)
\(P\cup Q \cup R=\{1,2,3,4,6,8,9,12,15,16,18,24\}\)

Draw a Venn diagram to represent sets \(P\)\(Q\) and \(R\), and shade the regions that represent \(P \cup Q\) and \(P\cup Q \cup R\).

\(P \cup Q\)
\(P \cup Q\cup R\)

 

 

The Complement of The Union of Sets
  • It is written as \((A \cup B)'\).
  • \((A \cup B)'\) read as " the complement of the union of sets of sets \(A\) and \(B\) ".
  • \((A \cup B)'\) refers to all the elements not in set \(A\) and set \(B\).
Example


Given the universal set, \(\xi=\{x:x\text{ is an integer},50\le x\le60\}\), set \(G=\{x:x\text{ is a prime number}\}\), set \(H=\{x:x\text{ is a multiple of 4}\}\) and set \(I=\{x:x\text{ is a multiple of 5}\}\), list all the elements and state the number of \((G\,\cup H)'\)\((G\,\cup I)'\)\((H\,\cup I)'\) and \((G\,\cup H\,\cup I)'\).

Solution:

\(\xi=\{50,51,52,53,54,55,56,57,58,59,60\}\)
\(G=\{53,59\}\)
\(H=\{52,56,60\}\)
\(I=\{50,55,60\}\)

\(n(G\,\cup H)'\) \((G\,\cup H)=\{52,53,56,59,60\}\)
\((G\,\cup H)'=\{50,51,54,55,57,58\}\)
\(n(G\,\cup H)'=6\)
\(n(G\,\cup I)'\) \((G\,\cup I)=\{50,53,55,59,60\}\)
\((G\,\cup I)'=\{51,52,54,56,57,58\}\)
\(n(G\,\cup I)'=6\)
\(n(H\,\cup I)'\) \((H\,\cup I)=\{50,52,55,56,60\}\)
\((H\,\cup I)'=\{51,53,54,57,58,59\}\)
\(n(H\,\cup I)'=6\)
\(n(G\,\cup H\,\cup I)'\) \((G\,\cup H\,\cup I)=\{50,52,53,55,56,59,60\}\)
\((G\,\cup H\,\cup I)'=\{51,54,57,58\}\)
\(n(G\,\cup H\,\cup I)'=4\)

 

 
The Complements of The Unions of Two or More Sets Using Venn Diagrams
Example


Three private travel agencies, \(A,B\) and \(C\) are chosen to organise the tourism exhibitions \(2020\) in Sarawak. Several divisions in Sarawak are chosen to hold the exhibition as follows:

\(\begin{aligned} \xi= \{&\text{Kapit}, \text{Miri}, \text{Bintulu},\text{Sibu}, \text{Limbang}, \text{Mukah}, \text{Kuching}, \text{Betong} \} \end{aligned}\)
\(\begin{aligned} A=\{&\text{Miri}, \text{Sibu}, \text{Kuching}, \text{Betong}\} \end{aligned}\)
\(\begin{aligned} B=\{&\text{Miri}, \text{Sibu}, \text{Kapit}, \text{Limbang}\} \end{aligned}\)
\(\begin{aligned} C=\{&\text{Miri}, \text{Betong}, \text{Kapit}, \text{Mukah}\} \end{aligned}\)

List all the elements and draw a Venn diagram to represent sets \(A\)\(B\) and \(C\), and shade the region that represents each of \(( A \cup B) '\)\(( B \cup C) '\) and \(( A \cup B \,\cup C) '\).

Solution:

\(( A \cup B) '\)
\(A\,\cup B=\{\text{Kapit, Miri, Sibu, Limbang, Kuching, Betong\}}\)
\((A\,\cup B)'=\{\text{Mukah, Bintulu\}}\)
 
\(( B \cup C) '\)
\(B\,\cup C=\{\text{Kapit, Miri, Sibu, Limbang, Betong, Mukah\}}\)
\((B\,\cup C)'=\{\text{Kuching, Bintulu\}}\)
 
\(( A \cup B \,\cup C) '\)
\(A\cup B\,\cup C=\{\text{Kapit, Miri, Sibu, Limbang, Mukah, Betong, Kuching\}}\)
\(( A \cup B \,\cup C) '=\{\text{Bintulu}\}\)

 

 
Solve Problem Involving the Union of Sets


A total of \(100\) adults are involved in a survey on their top choices of reading materials. \(40\) people choose newspapers, \(25\) people choose magazines, \(18\) people choose storybooks, \(8\) people choose both newspapers and magazines, \(7\) people choose both magazines and storybooks, \(5\) people choose both newspapers and storybooks, and \(3\) people choose all three types of reading materials.

How many people do not choose any of the reading materials.

Solution:

\(\xi=\{\text{total number of adults}\}\)
\(P=\{\text{newspapers}\}\)
\(Q=\{\text{magazines}\}\)
\(R=\{\text{storybooks}\}\)

  Total
Number of Adults \(100\)
Newspapers \(40\)
Magazines \(25\)
Storybooks \(18\)
Newspapers and Magazines \(8\)
Magazines and Storybooks \(7\)
Newspapers and Storybooks \(5\)
Newspapers, Magazines and Storybooks \(3\)
People Who Do Not Choose Any of the Reading Materials \(?\)
 
 

Newspapers only: \(40-5-3-2=30\)

Magazines only: \(25-5-3-4=13\)

Storybooks only: \(18-3-4-2=9\)

Total number of people who do not choose any of the reading materials: 
\(\quad n(A\,\cup B \,\cup C)'\\=100-5-3-4-2-30-13-9\\=34\)

 

Union of Sets

4.2 Union of Sets
 
Union of Sets
  • The union of sets \(P\) and \(Q\) is written using the symbol \(\cup\).
  • \(P \cup Q\) represents all the elements in set \(P\) or set \(Q\) or in both sets \(P\) and \(Q\).
 
The Union of Two or More Sets Using The Venn Diagram
Example


It is given that set \(P=\{\text{factors of 24}\}\), set \(Q=\{\text{multiples of 3 which are less than 20}\}\) and set \(R=\{\text{multiples of 4 which are less than 20}\}\).

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

\(P \cup Q\) \(P=\{1,2,3,4,6,8,12,24\}\)
\(Q=\{3,6,9,12,15,18\}\)
\(P\,\cup Q=\{1,2,3,4,6,8,9,12,15,18,24\}\)
\(P \cup R\) \(P=\{1,2,3,4,6,8,12,24\}\)
\(R=\{4,8,12,16\}\)
\(P\,\cup R=\{1,2,3,4,6,8,12,16,24\}\)
\(Q \cup R\) \(Q=\{3,6,9,12,15,18\}\)
\(R=\{4,8,12,16\}\)
\(Q\,\cup R=\{3,4,6,8,9,12,15,16,18\}\)
\(P\cup Q \cup R\) \(P=\{1,2,3,4,6,8,12,24\}\)
\(Q=\{3,6,9,12,15,18\}\)
\(R=\{4,8,12,16\}\)
\(P\cup Q \cup R=\{1,2,3,4,6,8,9,12,15,16,18,24\}\)

Draw a Venn diagram to represent sets \(P\)\(Q\) and \(R\), and shade the regions that represent \(P \cup Q\) and \(P\cup Q \cup R\).

\(P \cup Q\)
\(P \cup Q\cup R\)

 

 

The Complement of The Union of Sets
  • It is written as \((A \cup B)'\).
  • \((A \cup B)'\) read as " the complement of the union of sets of sets \(A\) and \(B\) ".
  • \((A \cup B)'\) refers to all the elements not in set \(A\) and set \(B\).
Example


Given the universal set, \(\xi=\{x:x\text{ is an integer},50\le x\le60\}\), set \(G=\{x:x\text{ is a prime number}\}\), set \(H=\{x:x\text{ is a multiple of 4}\}\) and set \(I=\{x:x\text{ is a multiple of 5}\}\), list all the elements and state the number of \((G\,\cup H)'\)\((G\,\cup I)'\)\((H\,\cup I)'\) and \((G\,\cup H\,\cup I)'\).

Solution:

\(\xi=\{50,51,52,53,54,55,56,57,58,59,60\}\)
\(G=\{53,59\}\)
\(H=\{52,56,60\}\)
\(I=\{50,55,60\}\)

\(n(G\,\cup H)'\) \((G\,\cup H)=\{52,53,56,59,60\}\)
\((G\,\cup H)'=\{50,51,54,55,57,58\}\)
\(n(G\,\cup H)'=6\)
\(n(G\,\cup I)'\) \((G\,\cup I)=\{50,53,55,59,60\}\)
\((G\,\cup I)'=\{51,52,54,56,57,58\}\)
\(n(G\,\cup I)'=6\)
\(n(H\,\cup I)'\) \((H\,\cup I)=\{50,52,55,56,60\}\)
\((H\,\cup I)'=\{51,53,54,57,58,59\}\)
\(n(H\,\cup I)'=6\)
\(n(G\,\cup H\,\cup I)'\) \((G\,\cup H\,\cup I)=\{50,52,53,55,56,59,60\}\)
\((G\,\cup H\,\cup I)'=\{51,54,57,58\}\)
\(n(G\,\cup H\,\cup I)'=4\)

 

 
The Complements of The Unions of Two or More Sets Using Venn Diagrams
Example


Three private travel agencies, \(A,B\) and \(C\) are chosen to organise the tourism exhibitions \(2020\) in Sarawak. Several divisions in Sarawak are chosen to hold the exhibition as follows:

\(\begin{aligned} \xi= \{&\text{Kapit}, \text{Miri}, \text{Bintulu},\text{Sibu}, \text{Limbang}, \text{Mukah}, \text{Kuching}, \text{Betong} \} \end{aligned}\)
\(\begin{aligned} A=\{&\text{Miri}, \text{Sibu}, \text{Kuching}, \text{Betong}\} \end{aligned}\)
\(\begin{aligned} B=\{&\text{Miri}, \text{Sibu}, \text{Kapit}, \text{Limbang}\} \end{aligned}\)
\(\begin{aligned} C=\{&\text{Miri}, \text{Betong}, \text{Kapit}, \text{Mukah}\} \end{aligned}\)

List all the elements and draw a Venn diagram to represent sets \(A\)\(B\) and \(C\), and shade the region that represents each of \(( A \cup B) '\)\(( B \cup C) '\) and \(( A \cup B \,\cup C) '\).

Solution:

\(( A \cup B) '\)
\(A\,\cup B=\{\text{Kapit, Miri, Sibu, Limbang, Kuching, Betong\}}\)
\((A\,\cup B)'=\{\text{Mukah, Bintulu\}}\)
 
\(( B \cup C) '\)
\(B\,\cup C=\{\text{Kapit, Miri, Sibu, Limbang, Betong, Mukah\}}\)
\((B\,\cup C)'=\{\text{Kuching, Bintulu\}}\)
 
\(( A \cup B \,\cup C) '\)
\(A\cup B\,\cup C=\{\text{Kapit, Miri, Sibu, Limbang, Mukah, Betong, Kuching\}}\)
\(( A \cup B \,\cup C) '=\{\text{Bintulu}\}\)

 

 
Solve Problem Involving the Union of Sets


A total of \(100\) adults are involved in a survey on their top choices of reading materials. \(40\) people choose newspapers, \(25\) people choose magazines, \(18\) people choose storybooks, \(8\) people choose both newspapers and magazines, \(7\) people choose both magazines and storybooks, \(5\) people choose both newspapers and storybooks, and \(3\) people choose all three types of reading materials.

How many people do not choose any of the reading materials.

Solution:

\(\xi=\{\text{total number of adults}\}\)
\(P=\{\text{newspapers}\}\)
\(Q=\{\text{magazines}\}\)
\(R=\{\text{storybooks}\}\)

  Total
Number of Adults \(100\)
Newspapers \(40\)
Magazines \(25\)
Storybooks \(18\)
Newspapers and Magazines \(8\)
Magazines and Storybooks \(7\)
Newspapers and Storybooks \(5\)
Newspapers, Magazines and Storybooks \(3\)
People Who Do Not Choose Any of the Reading Materials \(?\)
 
 

Newspapers only: \(40-5-3-2=30\)

Magazines only: \(25-5-3-4=13\)

Storybooks only: \(18-3-4-2=9\)

Total number of people who do not choose any of the reading materials: 
\(\quad n(A\,\cup B \,\cup C)'\\=100-5-3-4-2-30-13-9\\=34\)