Set Operations

Set Operations

Objective

• Perform the basic operations on sets.

Open Ended

What is a set?

Definition of a Set

• a collection of objects called elements or members of the set

• enclosed in braces { }

• denoted by capital letters

Examples

•  $A=\left\{a,\ e,\ i,\ o,\ u\right\}$  

•  $B=\left\{2,\ 4,\ 6,\ 8,\ 10\right\}$  

•  $C=\left\{grade\ 7\ students\ of\ PSHS\ -\ CARC\right\}$  

Set Operations

UNION

 $A\ \cup\ B\ =\ \left\{x\ :\ x\ \in\ A\ or\ x\ \in\ B\right\}$  

Examples

A = {1, 3, 5}

B = {2, 3, 5, 7}

C = {2, 4, 6, 8}

D = {9, 10}

Multiple Choice

 $A\ =\ \left\{1,\ 3,\ 5,\ 7,\ 9\right\}$   $B\ =\ \left\{2,\ 4,\ 6,\ 8,\ 10\right\}$  

 $A\ \cup\ B\ =\ ?$  

•  $A\cup B=\left\{1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10\right\}$  

•  $A\cup B=\left\{\right\}$  

•  $A\cup B=\left\{x\in N\ :\ x<10\right\}$  

Multiple Choice

 $A=\left\{1,\ 3,\ 5\right\}$  
 $B=\left\{2,\ 3,\ 4\right\}$  
 $C=\left\{1,\ 6\right\}$  
 $A\cup B\cup C=?$  

•  $\left\{1,\ 2,\ 3,\ 4,\ 5,\ 6\right\}$  

•  $\left\{1,\ 3\right\}$  

•  $\left\{2,\ 4,\ 5,\ 6\right\}$  

Set Operations

 INTERSECTION

 $A\ \cap\ B\ =\ \left\{x\ :\ x\ \in\ A\ and\ x\ \in\ B\right\}$ 

Examples

A = {1, 3, 5, 7, 9}

B = {2, 4, 6, 8, 10}

C = {3, 6, 7}

D = {1, 8, 10}

Multiple Choice

 $A=\left\{1,\ 2,\ 3,\ 4\right\}$  
 $B=\left\{3,\ 4,\ 5,\ 6,\ 7\right\}$  
 $A\cap B=?$  

•  $A\cap B=\left\{3,\ 4\right\}$  

•  $A\cap B=\left\{1,\ 2,\ 5,\ 6,\ 7\right\}$  

•  $A\cap B=\phi$  

Multiple Choice

 $A=\left\{3,\ 6,\ 9,\ 12\right\}$  
 $B=\left\{2,\ 4,\ 6,\ 8\right\}$  
 $C=\left\{2,\ 3,\ 8\right\}$  
 $A\cap B\cap C=?$  

•  $A\cap B\cap C=\left\{\right\}$  

•  $A\cap B\cap C=\left\{\phi\right\}$  

•  $A\cap B\cap C=\left\{2,\ 3,\ 6,\ 8\right\}$  

Set Operation

COMPLEMENT


 $A'\ =\ \left\{x\ :\ x\ \notin\ A\right\}$  or  $A'\ =\ \left\{x\ :\ x\ \in\ U\ and\ x\ \notin\ A\right\}$  

Examples

U = {1, 2, 3, ... , 10}

A = {1, 3, 4, 7}

B = {2, 5, 8, 10}

C = {2, 4, 6, 8, 10}

D = {1, 3, 5, 7, 9}

Multiple Choice

 $U=\left\{1,\ 2,\ 3,\ ...\ ,\ 10\right\}$  
 $A=\left\{1,\ 2,\ 3,\ 4\right\}$  
 $A'=?$  

•  $A'=\left\{5,\ 6,\ 7,\ 8,\ 9,\ 10\right\}$  

•  $A'=\left\{1,\ 2,\ 3,\ 4\right\}$  

•  $A'=\phi$  

Multiple Choice

 $U=\left\{1,\ 2,\ 3,\ 4,\ ...\ ,\ 10\right\}$  
 $B=\left\{2,\ 4,\ 8,\ 10\right\}$  
 $B'=?$  

•  $B'=\left\{1,\ 3,\ 5,\ 6,\ 9\right\}$  

•  $B'\ =\ \left\{1,\ 3,\ 5,\ 7,\ 9\right\}$  

•  $B'=\left\{1,\ 3,\ 5,\ 6,\ 7,\ 9\right\}$  

Set Operation

DIFFERENCE

 $A\ -B\ =\ \left\{x\ :\ x\ \in\ A\ and\ x\ \notin\ B\right\}$  

Examples

A = {1, 2, 3}

B = {2, 3, 5, 7}

C = {3, 7, 8, 11}

D = {1, 2, 8, 15}

Multiple Choice

 $A=\left\{5,\ 10,\ 15,\ 20,\ 25\right\}$  
 $B=\left\{10,\ 20,\ 30,\ 40\right\}$  
 $A-B=?$  

•  $A-B=\left\{5,\ 15,\ 25\right\}$  

•  $A-B=\left\{30,\ 40\right\}$  

•  $A-B=\left\{5,\ 15,\ 30,\ 40\right\}$  

Multiple Choice

 $A=\left\{even\ numbers\right\}$  
 $B=\left\{odd\ numbers\right\}$  
 $B-A=?$  

•  $B-A=\phi$  

•  $B-A=\left\{even\ numbers\right\}$  

•  $B-A=\left\{odd\ numbers\right\}$  

SET OPERATIONS

• UNION:  $A\cup B=\left\{x\ :\ x\in A\ or\ x\in B\right\}$  

• INTERSECTION:  $A\cap B=\left\{x\ :\ x\in A\ and\ x\in B\right\}$  

• COMPLEMENT:  $A'=\left\{x\ :\ x\in U\ and\ x\notin A\right\}$  

• DIFFERENCE:  $A-B=\left\{x\ :\ x\in A\ and\ x\notin B\right\}$  

References

K to 12 grade 7 learning module in mathematics (Q1 – Q2). (2014, March 11). Retrieved August 26, 2018 from http://www.slideshare.net/mobile/lhoralight/k-to-12-grade-7-learning-material-in-mathematics-q1q2

 

Sets and Venn diagrams (Number and Algebra: Module 1). (2011, June). Retrieved August 26, 2018 from http://www.amsi.org.au/teacher_modules/pdfs/Sets_and_venn_diagrams.pdf

 

Velleman, D. (2006). How to prove it: A structured approach second edition. UK: Cambridge University Press.