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Why is a set of positive integers not a group under the operation of addition?
The set of positive integers does not contain the additive inverses of all but the identity. It is, therefore, not a group.
Why the set of odd integers under addition is not a group?
Because the set is not closed under addition. If x and y are odd, then x + y is not odd.
Why are odd integers closed under multiplication but not under addition?
The numbers are not closed under addition because whole numbers, even integers, and natural numbers are closed.
Examples to prove the set of integers is a group with respect to addition?
In order to be a group with respect to addition, the integers must satisfy the following axioms: 1) Closure under addition 2) Associativity of addition 3) Contains the additive identity 4) Contains the additive inverses 1) The integers are closed under addition since the sum of any two integers is an integer. 2) The integers are associative with respect to addition since (a+b)+c = a+(b+c) for any integers a, b, and c. 3) The integer 0 is the additive identity since z+0 = 0+z = z for any integer z. 4) Each integer n has an additive inverse, namely -n since n+(-n) = -n+n = 0.
ARe odd integers not closed under addition?
That is correct, the set is not closed.
