Tell whether the set is closed under the operation. If it is not closed give an example that shows that the set is not closed under the operation.
10. Positive irrational numbers; division
11. Negative rational numbers; multiplication
12. Negative integers; addition
13. Positive integers; subtraction
yes: 12
no: 10,11,13
√2/√2 = 1
-2/3 * -3/2 = 1
3-2 = -1
just tell me if ,Under the set of rational numbers is -9 + 12 closed?
Give one counter example for each that the set of irrational number is not closed under the operations of addition, substraction, multiplication and division
To determine whether a set is closed under an operation, we need to examine whether performing that operation on elements of the set always gives us a result that is also in the set.
Let's consider each set and operation:
10. Positive irrational numbers and division:
To check if the set of positive irrational numbers is closed under division, we need to divide any two positive irrational numbers and see if the result is also a positive irrational number.
Example:
Take the numbers √2 and √3, both of which are positive irrational numbers. When we divide √2 by √3, we get (√2)/(√3) = (√2/√3) = (√2√3)/(√3√3) = (√6)/3. Since √6 is also an irrational number and the result is positive, we can conclude that the set of positive irrational numbers is closed under division.
11. Negative rational numbers and multiplication:
To check if the set of negative rational numbers is closed under multiplication, we need to multiply any two negative rational numbers and see if the result is also a negative rational number.
Example:
Consider the numbers -1/2 and -3/4, both of which are negative rational numbers. When we multiply -1/2 by -3/4, we get (-1/2)(-3/4) = 3/8. Since 3/8 is a positive rational number, not a negative rational number, we can conclude that the set of negative rational numbers is not closed under multiplication.
12. Negative integers and addition:
To check if the set of negative integers is closed under addition, we need to add any two negative integers and see if the result is also a negative integer.
Example:
Take the negative integers -5 and -8. When we add -5 to -8, we get -13. Since -13 is also a negative integer, we can conclude that the set of negative integers is closed under addition.
13. Positive integers and subtraction:
To check if the set of positive integers is closed under subtraction, we need to subtract one positive integer from another and see if the result is also a positive integer.
Example:
Consider the positive integers 7 and 12. When we subtract 12 from 7, we get 7 - 12 = -5. Since -5 is not a positive integer, but a negative integer, we can conclude that the set of positive integers is not closed under subtraction.
In summary:
- The set of positive irrational numbers is closed under division.
- The set of negative rational numbers is not closed under multiplication.
- The set of negative integers is closed under addition.
- The set of positive integers is not closed under subtraction.