All about sets

Question

Why is that the elements/answer of set A is {1/3, -2} from the notation {x:(3x-1)(x+2)=0

 

Please explain 

 

AND

What is the cardinality of A = {}?

Should we consider it even if it is not enclosed in brackets?

 

Answer 1

{x:(3x-1)(x+2)=0} means "the set of all x such that (3x-1)(x+2)=0".  So what values of x make that statement true?  if x is 1/3, then 3x-1 = 0, so the entire expression is 0.  And if x is -2, then x+2 = 0, so again the entire expression is 0.
And there are no other values of x that make the expression 0. 

 

Therefore, the set includes only the numbers 1/3 and -2, so the solution set is {1/3, -2}.

 

 

{} is the null set, which has no elements.  Therefore, its cardinality is 0.