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Chapter 1: Problem 6

Is the set of all natural numbers from 1 to 10 a closed system under addition?

Short Answer

Expert verified
The set of all natural numbers from 1 to 10 (S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}) is not a closed system under addition, as there are sums like 1 + 10 = 11, 3 + 9 = 12, and 5 + 10 = 15 that fall outside the original set S.

Step by step solution

01

Define the addition operation

Consider the addition operation, which is denoted by (+). We need to check if the sum of any two elements in the set S is also in the set S.
02

Create a table of sums

We can create a table of sums, where we can check if the sum of any two elements within the set S falls within the set S. The table will contain rows and columns representing the elements in set S: + | 1 2 3 4 5 6 7 8 9 10 --------------------------------- 1 | 2 3 4 5 6 7 8 9 10 11 2 | 3 4 5 6 7 8 9 10 11 12 3 | 4 5 6 7 8 9 10 11 12 13 4 | 5 6 7 8 9 10 11 12 13 14 5 | 6 7 8 9 10 11 12 13 14 15 6 | 7 8 9 10 11 12 13 14 15 16 7 | 8 9 10 11 12 13 14 15 16 17 8 | 9 10 11 12 13 14 15 16 17 18 9 | 10 11 12 13 14 15 16 17 18 19 10 | 11 12 13 14 15 16 17 18 19 20
03

Analyze the table

From the table, we can see that there are sums that fall outside the original set S. For example, the sum 1 + 10 = 11, 3 + 9 = 12, and 5 + 10 = 15 are not within the original set S.
04

Determine if the set is closed under addition

Since there are sums that fall outside the set S, the set of all natural numbers from 1 to 10 is not a closed system under addition.

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