Why the elements from 23 30 has no valenxy
Answers
Problems like this are great because they are easily understood conceptually.
Assume that set A has 30 element and set B has 40 elements. Now if you were to list all of the elements, including duplicates, you would get 70 elements. By assumption, their union-- the collection of all distinct elements-- is equal to 45. Thus there are 70 - 45 = 25 non-distinct elements. That is, there are 25 elements in their intersection.
Formally, we say that the order of the intersection of two sets is equal to the sum of the orders of the two sets less the order of their union. Or:
|(A∩B)| = |A| + |B| - |A∪B| where |x| denotes the order, or size, of the set x.
However, from above you can see that this follows directly from the concepts. It is important that we teach students the concepts, and not merely the manipulation of symbols.
As an aside, I think that you are merely too quickly glossing over 70-45 to arrive at 35; happens to the best of us.