Help with this "Elementary School Math Problem?"?

Question

I don't understand how this is elementary school math. I'm very upset because I just don't get it. The question is:

Which of the following sets are closed under addition? Why or why not? There might be more than 1 answer.

a. Whole numbers less than 18.
b. {16,24,32,40,...}
c. {0,9,18,27,...}
d. {3}
e. {1,9,17,25,...}

Can someone please explain this nonsense?

? · Accepted Answer

Closed under addition means the sum of any two numbers is in the given set

a)  { 1, 2, 3, 4,    ...   17}

3  +  17  =  20,  is not in the set.  So not closed under addition

b. {16,24,32,40,...}

These are multiples of 8,   8(n + 1).  The sum of any two is also a multiple of 8

So  the set is closed under addition

c. {0,9,18,27,...}

These are a multiple of 9,   9(n - 1)

The sum of any two numbers of set is also a multiple of 9.
 
        The set is closed under addition

d. {3} 
     
      3 + 3 = 6,  not in the set,  so not closed

e. {1,9,17,25,...}

1 + 9  =  10,  10 is not in the set.  Therefore not closed under addition

? · Answer

closed under addition means that if you add any two things in the set your answer is also in the set

Using ordinary addition you can eliminate any finite set except the set {0}. Add the largest number to itself and find a number outside the set. So scratch (a) & (d). 
e) is 1+a multiple of 8 so no (1+9=10, it fails) so no
b) is multiples of 8 so yes
c) is multiples of 9 so yes

? · Answer

"Closed under addition" means that if you add any two numbers in the set, the sum is also in the set.

d. and e. are not closed. 
3+3=6 is not in d.
1+9=10 is not in e.

For the others, it's easy to verify closure.

This could be covered and understood perhaps in 6th grade and up, once students understand arithmetic pretty well.

Help with this "Elementary School Math Problem?"?

3 Answers