Proving the Sum of Additive Groups Z: (3/7)Z + (11/2)Z = (1/14)Z

In summary, we have two sets, (3/7)Z and (11/2)Z, and we need to prove that they are equal to the set (1/14)Z. By definition, (3/7)Z+(11/2)Z can be rewritten as {(6k + 77m)/14 : k,m € Z}. While it is easy to show that this set is a subset of (1/14)Z, the converse is more difficult. However, by choosing specific values for k and m, such as k=13 and m=-1, we can show that 1/14 is also in the group 3/7Z+11/2Z. Therefore, all
  • #1
bedi
81
0
Z is the set of integers. Prove that (3/7)Z + (11/2)Z = (1/14)Z

Attempt:

By definition,
(3/7)Z+(11/2)Z={3k/7 + 11m/2 : k,m € Z} = {(6k + 77m)/14 : k,m € Z}.

Showing that 3/7Z+11/2Z is a subset of 1/14 Z is easy but I can't prove the converse. Can't show that whatever n€1/14Z I take satisfies that n€3/7Z+11/2Z.
 
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  • #2
bedi said:
Z is the set of integers. Prove that (3/7)Z + (11/2)Z = (1/14)Z

Attempt:

By definition,
(3/7)Z+(11/2)Z={3k/7 + 11m/2 : k,m € Z} = {(6k + 77m)/14 : k,m € Z}.

Showing that 3/7Z+11/2Z is a subset of 1/14 Z is easy but I can't prove the converse. Can't show that whatever n€1/14Z I take satisfies that n€3/7Z+11/2Z.

If you could show 1/14 is in the group 3/7Z+11/2Z, then all multiples of 1/14 would also be in the group, right?
 
  • #3
Yes! If I take k=13 and m=-1 then 1/14 is in the group 3/7Z+11/2Z. Thank you very much.
 

1. What is the sum of additive groups?

The sum of additive groups refers to the mathematical operation of combining two or more groups together to form a larger group. This can be represented using the "+" symbol and follows the rules of addition such as commutativity and associativity.

2. How is the sum of additive groups calculated?

To calculate the sum of additive groups, you simply add together all the elements in each group. For example, if you have two groups A = {1, 2, 3} and B = {4, 5, 6}, the sum of A and B would be {1+4, 2+5, 3+6} which equals {5, 7, 9}.

3. Can the sum of additive groups be applied to any type of group?

Yes, the sum of additive groups can be applied to any type of group as long as the group follows the rules of addition. This includes groups in mathematics, chemistry, and even in social sciences.

4. How is the sum of additive groups different from the sum of multiplicative groups?

The sum of additive groups and the sum of multiplicative groups are two different mathematical operations. The sum of additive groups involves adding elements together, while the sum of multiplicative groups involves multiplying elements together. Additionally, the rules for each operation are different.

5. What are some real-life applications of the sum of additive groups?

The sum of additive groups has various real-life applications, such as in calculating the total cost of items in a shopping cart, determining the total weight of multiple objects, and finding the total distance traveled by a vehicle on a trip. It is also used in fields such as economics, physics, and engineering for various calculations and analyses.

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