Answer and Step-by-step explanation:
1-A= {all even integers} under multiplication?:
Sol: The set of positive integers under multiplication would not form because this set does not contain inverses. And no identity exists in this set. And in the case of negative even integers under multiplication, when we multiply a negative number to another negative number it becomes a positive even integer.
2-B= {all positive even integers} under averaging?:
The average of the integers from 1 to n is (n+1)/2.
Because there is only even numbers so, the formula has to doubled, this formula is used for sequences that increase by 1. So n(n+1)
There are only 1009 even integers between 2 to 2018.
2+2018=2020 and 2020/2=1010
3-C= {all odd integers} under addition:
Sol: the set of odd integers under addition does not form a set because the set of odd integers is not a closed set. Because addition of two odd numbers gives us an even number. And the other reason is that the identity element does not exist in this set. Identity for addition is zero (0) which is an even number.
4-D= {all odd integers} under multiplication:
Sol: The set of odd integers say F is not a group under multiplication (.) because the inverse of odd integers does not exist. For example an element m is belonging to odd integer group say F. its inverse (1/m) does not exist in this set. Sometimes it does not have identity.
5- E= {all prime natural numbers) under addition:
Sol: All prime natural numbers are not set under addition because when we add two prime numbers we will not get always a prime number. For example 7+3= 10
7 and 3 are prime numbers but 10 is not a prime number.
6-F= {all composite integers} under addition:
Sol: no, not all composite number under addition form a set because when we add two composite number 15 + 14 = 29, 29 is a prime number.
7-G= {all composite integers} under multiplication:
Sol: yes all composite integers form a set under multiplication because when we multiply any integer to another integer it gives composite numbers. When we multiply prime numbers it gives a composite number.
