Question
Which of the following sets is closed under the operation of addition?
Question 8Answer
a.
Whole numbers
b.
Odd numbers
c.
Natural numbers
d.
Integers
Question 8Answer
a.
Whole numbers
b.
Odd numbers
c.
Natural numbers
d.
Integers
Answers
Answered by
GPT-4o mini
To determine which of the provided sets is closed under addition, we need to analyze each set to see if the sum of any two elements within the set also belongs to the same set.
a. **Whole numbers**: The whole numbers include 0, 1, 2, 3, ..., and if we add any two whole numbers, the result is always a whole number. Therefore, this set is closed under addition.
b. **Odd numbers**: Odd numbers are of the form 1, 3, 5, 7, ... If we add two odd numbers (e.g., 3 + 5 = 8), the result is an even number, which is not in the set of odd numbers. Therefore, this set is not closed under addition.
c. **Natural numbers**: The natural numbers are typically defined as 1, 2, 3, ... If we add two natural numbers (e.g., 2 + 3 = 5), the result is still a natural number. Thus, this set is closed under addition.
d. **Integers**: The integers include all whole numbers and their negatives, ..., -3, -2, -1, 0, 1, 2, 3, ... If we add any two integers (e.g., -1 + 2 = 1), the result is also an integer. Hence, this set is closed under addition.
Based on this analysis, the sets that are closed under addition are:
- Whole numbers
- Natural numbers
- Integers
Since the question asks for which set (among the options provided) is closed under addition, the answer would be **a. Whole numbers**, **c. Natural numbers**, and **d. Integers**. If only one answer must be chosen, then the **whole numbers** (option a) are the most inclusive choice.
a. **Whole numbers**: The whole numbers include 0, 1, 2, 3, ..., and if we add any two whole numbers, the result is always a whole number. Therefore, this set is closed under addition.
b. **Odd numbers**: Odd numbers are of the form 1, 3, 5, 7, ... If we add two odd numbers (e.g., 3 + 5 = 8), the result is an even number, which is not in the set of odd numbers. Therefore, this set is not closed under addition.
c. **Natural numbers**: The natural numbers are typically defined as 1, 2, 3, ... If we add two natural numbers (e.g., 2 + 3 = 5), the result is still a natural number. Thus, this set is closed under addition.
d. **Integers**: The integers include all whole numbers and their negatives, ..., -3, -2, -1, 0, 1, 2, 3, ... If we add any two integers (e.g., -1 + 2 = 1), the result is also an integer. Hence, this set is closed under addition.
Based on this analysis, the sets that are closed under addition are:
- Whole numbers
- Natural numbers
- Integers
Since the question asks for which set (among the options provided) is closed under addition, the answer would be **a. Whole numbers**, **c. Natural numbers**, and **d. Integers**. If only one answer must be chosen, then the **whole numbers** (option a) are the most inclusive choice.