Determine if each of the following sets is closed under addition. $$ \begin{array}{ll} ext { a) } L=\{0\} & ext { c) } N=\{6,9,11\} \ ext { b) } S=\{0,6,12,18,24 \ldots\} & ext { d) }\{x \mid x \in ext { integers, } x10\}\end{array} $$ $$ \begin{array}{l} ext { a) } L=\{0\} ext { is closed under addition. } \ ext { True } \ ext { False }\end{array} $$

Question

Determine if each of the following sets is closed under addition. $$ \begin{array}{ll}	ext { a) } L=\{0\} & 	ext { c) } N=\{6,9,11\} \ 	ext { b) } S=\{0,6,12,18,24 \ldots\} & 	ext { d) }\{x \mid x \in 	ext { integers, } x>10\}\end{array} $$ $$ \begin{array}{l}	ext { a) } L=\{0\} 	ext { is closed under addition. } \ 	ext { True } \ 	ext { False }\end{array} $$

UpStudy AI · Accepted Answer

To determine if each set is closed under addition, we need to check if the sum of any two elements from the set is also an element of the same set. Let's analyze each set one by one.

### a) $$ L = \{0\} $$

- The only element in this set is $$0$$.
- Adding $$0 + 0 = 0$$, which is in the set.

**Conclusion**: $$ L $$ is closed under addition.  
**Answer**: True

### b) $$ S = \{0, 6, 12, 18, 24, \ldots\} $$

- This set consists of all non-negative multiples of $$6$$.
- Let's check the addition of two elements:
  - $$6 + 6 = 12$$ (in $$S$$)
  - $$6 + 12 = 18$$ (in $$S$$)
  - $$12 + 12 = 24$$ (in $$S$$)
  - $$0 + 6 = 6$$ (in $$S$$)
  - In general, if $$a$$ and $$b$$ are multiples of $$6$$, then $$a + b$$ is also a multiple of $$6$$.

**Conclusion**: $$ S $$ is closed under addition.  
**Answer**: True

### c) $$ N = \{6, 9, 11\} $$

- Let's check the addition of elements:
  - $$6 + 6 = 12$$ (not in $$N$$)
  - $$6 + 9 = 15$$ (not in $$N$$)
  - $$6 + 11 = 17$$ (not in $$N$$)
  - $$9 + 9 = 18$$ (not in $$N$$)
  - $$9 + 11 = 20$$ (not in $$N$$)
  - $$11 + 11 = 22$$ (not in $$N$$)

Since the sum of two elements from $$N$$ can produce results that are not in $$N$$, it is not closed under addition.

**Conclusion**: $$ N $$ is not closed under addition.  
**Answer**: False

### d) $$ \{x \mid x \in 	ext{ integers, } x > 10\} $$

- Let's check the addition of two elements:
  - If $$a > 10$$ and $$b > 10$$, then $$a + b > 20$$, which is also greater than $$10$$.
  
Since the sum of any two integers greater than $$10$$ is also greater than $$10$$, this set is closed under addition.

**Conclusion**: The set is closed under addition.  
**Answer**: True

### Summary of Answers:
- a) True
- b) True
- c) False
- d) True
- a) True - b) True - c) False - d) True

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