Chapter 2: Operation of Sets

 

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Mathematics for Biotechnology and Data Science 

Chapter 1: Set Theory 

Chapter 2: Operation of Sets 

Chapter 3: Application of Set Theory 

Chapter 4: Venn Diagram  

Chapter 5: Relation  

Chapter 6: MAPPING OR FUNCTIONS 

Chapter 8: Inverse of a Function 

Chapter 2: Operation of Sets 

2.1. Union of Sets: Let A and B be two sets. The union of A and B is the set of all those elements which belong either to A or to B or to both A and B.

We shall use notation  (reads as “A union B”) to denote the union of A and B.

Thus,.

Clearly, .

and,.

It is evident form the definition that

Example 1: If A = {1, 2, 3} and B = {1, 3, 5, 7}, then .

Example 2: If  and  then

 

2.2. Intersection of Sets: Let A and B be two sets. The intersection of A and B is the set of all those elements that belong to both A and B.  

We shall use notation  (reads as “A intersection B”) to denote the intersection of A and B.

Thus,.

Clearly, .

Example 3: If A = {1, 2, 3, 4, 5} and B = {1, 3, 9, 12}, then.

Example 4: If  and  then

 

            

Example 5: If  and  then find .

Solution: We have,

 x is multiple of 3 and x is a multiple of 4

 x is a multiple of 3 and 4 both

 x is multiple of 12

 

Hence, .

2.3. Complement Sets: Let U be the universal set and let A be a set such that . Then the complement of A with respect to U is denoted by  or  or  and defined the set of all those elements of U which are not in A.

Thus .

Clearly  

Example 7: Let the set of natural numbers N = {1, 2, 3, 4,…} be the universal set and let A = {2, 4, 6, 8,…} then .

Example 8: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 3, 5, 7, 9} then .

EXERCISE

Q:1 If A and B are two sets such that  then find (i)  (ii) Ans (1) A (2) B

Q:2 If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14} then Find (1)    (2)    (3)    (4)    (5)    (6)    (7) (8)    (9)    (10) Ans (1) {1, 2, 3, 4, 5, 6, 7, 8}  (2) {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}  (3) {4, 5, 6, 7, 8, 9, 10, 11} (4) {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}  (5) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}  (6) {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14}  (7) {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}  (8) {4, 5}  (9)   (10) {4, 5,10,11}

Q:3 Let  , B , C and   then Find (1)    (2)    (3)    (4) B   (5) B   (6) C Ans (1) B (2) C (3) D (4)  (5) {2} (6)

Q:4 If A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12 , 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20} then Find (1) A B    (2) A C  (3) AD  (4) B A  (5) C A  (6) D A Ans (1) {12} (2) {2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 21} (3){3, 5, 6, 10, 12, 15, 18, 20, 21} (4) {3, 6, 12, 15, 18, 21, 4, 8, 16, 20} (5){6, 12} (6) {15}

Q:5 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6} then Find (1)    (2)   (3)    (4)   (5)   Ans (1) {5, 6, 7, 8, 9} (2) {1, 3, 5, 7, 9} (3){ 1, 2, 5, 6, 7, 8, 9} (4) {5, 7, 9} (5) A

Q:6 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7} then verify that (1) and  (2)

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