Let A be the set of all composites less than 10, and B be the set of positive even integers less than 10. How many different sums of the form a + b are possible if a is in A and b is in B?

1 Answer
MathFact-orials.blogspot.com
Aug 26, 2017

16 different forms of #a+b#. 10 unique sums.

Explanation:

The set #bb(A)#

A composite is a number that can be divided evenly by a smaller number other than 1. For instance, 9 is composite #(9/3=3)# but 7 is not (another way of saying this is a composite number is not prime). This all means that the set #A# consists of:

#A={4,6,8,9}#

The set #bb(B)#

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

We're now asked for the number of different sums in the form of #a+b# where #a in A, b in B#.

In one reading of this problem, I'd say there are 16 different forms of #a+b# (with things like #4+6# being different than #6+4#).

However, if read as "How many unique sums are there?", perhaps the easiest way to find that is to table it out. I'll label the #a# with #color(red)("red")# and #b# with #color(blue)("blue")#:

#(("",color(blue)2,color(blue)4,color(blue)6,color(blue)8),(color(red)4,6,8,10,12),(color(red)6,8,10,12,14),(color(red)8,10,12,14,16),(color(red)9,11,13,15,17))#

And so there are 10 unique sums: #6, 8, 10, 11, 12, 13, 14, 15, 16, 17#

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