Question

a.      Let A and B be sets as follows, A= {1,2,3,5,7} and B = {1,5,6,8}. Find:

i.            A∩B

ii.          | A∩B|

iii.        P(A∩B)

iv.        P(A) and P(B)

v.          | P(A) | and | P(B) |

vi.        The Cartesian product of A and B. vii.         |A|, |B| and hence find |A x B|  

Answer 1

i.            $A\cap B=\{1,5\}$

ii.         $A\cup B=\{1,2,3,5,6,7,8\}$

$|A\cup B=|=7$

iii.       $P(A\cap B)=P(\{1,5\})=\{\empty,\{1\}, \{5\},\{1,5\}\}$  P(A∩B)

iv.        

$$P(A)=P(\{1,2,3,5, 7\})$$ $$=\{\empty,\{1\}, \{2\},\{3\},\{5\},\{7\},\{1,2\},\{1,3\},\{1,5\},$$

$$\{1,7\},\{2,3\},\{2,5\},\{2,7\},\{3,5\},\{3,7\},\{5,7\},$$

$$\{1, 2,3\},\{1, 2,5\},\{1, 2,7\},\{1, 3,5\},\{1, 3,7\},$$

$$\{1, 5,7\},\{2, 3,5\},\{2, 3,7\},\{2, 5,7\},\{3, 5,7\},$$

$$\{1, 2,3,5\},\{1, 2,3,7\},\{1, 2,5,7\},\{1, 3,5, 7\},$$

$$\{2, 3,5,7\},\{1, 2,3,5,7\}\}$$

 

$$P(B)=P(\{1,5, 6,8\})$$ $$=\{\empty,\{1\}, \{5\},\{6\},\{8\},\{1,5\},\{1,6\},\{1,8\},$$

$$\{5,6\},\{5,8\},\{6,8\},\{1,5, 6\},\{1,5, 8\},\{1,6,8\},$$

$$\{5, 6,8\},\{1, 5,6,8\}\}$$

v.    

$$|P(A)|=2^5=32$$

      

$$|P(B)|=2^4=16$$

vi.        

$$A\times B=\{(1,1), (1,5), (1,6), (1,8),$$ $$(2,1), (2,5),(2,6), (2,8),(3,1), (3,5), (3,6), (3,8),$$

$$(5,1), (5,5), (5,6),(5,8),(7,1), (7,5), (7,6),(7,8)\}$$

vii.     

$$|A|=5, |B|=4$$

   

$$|A\times B|=5(4)=20$$