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The set of positive integers does not contain the additive inverses of all but the identity. It is, therefore, not a group.

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Q: Why is a set of positive integers not a group under the operation of addition?
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Is the set of all negative integers a group under addition?

no


What groups are subsets of integers?

Integer Subsets: Group 1 = Negative integers: {... -3, -2, -1} Group 2 = neither negative nor positive integer: {0} Group 3 = Positive integers: {1, 2, 3 ...} Group 4 = Whole numbers: {0, 1, 2, 3 ...} Group 5 = Natural (counting) numbers: {1, 2, 3 ...} Note: Integers = {... -3, -2, -1, 0, 1, 2, 3 ...} In addition, there are other (infinitely (uncountable infinity) many) other subsets. For example, there is the set of even integers. There is also the subset {5,7}.


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.


0 belongs to what family of real numbers?

0 belongs to the reals. It is a member of the irrationals, the rationals. It is also a member of the integers; It is a member (the identity) of the group of even integers, 3*integers, 4*integers etc with respect to addition.


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.

Related questions

Is the set of positive integers a commutative group under the operation of addition?

No. It is not a group.


Give two reason for the set of odd integers is not a group?

Assuming that the question is in the context of the operation "addition", The set of odd numbers is not closed under addition. That is to say, if x and y are members of the set (x and y are odd) then x+y not odd and so not a member of the set. There is no identity element in the group such that x+i = i+x = x for all x in the group. The identity element under addition of integers is zero which is not a member of the set of odd numbers.


Is the set of negative integers a group under addition?

Is the set of negative interferes a group under addition? Explain,


Is the set of all negative integers a group under addition?

no


Does the set of even integers form a group under the operation of multiplication?

No. The inverses do not belong to the group.


What groups are subsets of integers?

Integer Subsets: Group 1 = Negative integers: {... -3, -2, -1} Group 2 = neither negative nor positive integer: {0} Group 3 = Positive integers: {1, 2, 3 ...} Group 4 = Whole numbers: {0, 1, 2, 3 ...} Group 5 = Natural (counting) numbers: {1, 2, 3 ...} Note: Integers = {... -3, -2, -1, 0, 1, 2, 3 ...} In addition, there are other (infinitely (uncountable infinity) many) other subsets. For example, there is the set of even integers. There is also the subset {5,7}.


What are the rules of integers?

To start with, the set of integers is a Group. This means that it is a set of elements (numbers) with a binary operation (addition) that combines any two elements in the set to form a third element. This Group, Z, satisfies four axioms: closure, associativity, identity and invertibility. that is, if x , y and z are integers, thenx + y is an integer (closure).(x + y) + z = x + (y + z) (associativity)there is an integer, denoted by 0, such that 0 + x = x + 0 = xthere is an integer, denoted by -x, such that x + (-x) = (-x) + x = 0.In addition, it is a Ring. A ring is an Abelian group (that is, addition is commutative: x + y = y + x) and it has a second binary operation (multiplication) that is defined on its elements. This second operation satisfies the axioms of closure, associativity and identity. It is also distributive over the first operation. That is,x*(y + z) = x*y + x*z


Example of group is an abelian group?

The set of integers, under addition.


0 belongs to what family of real numbers?

0 belongs to the reals. It is a member of the irrationals, the rationals. It is also a member of the integers; It is a member (the identity) of the group of even integers, 3*integers, 4*integers etc with respect to addition.


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.


What is finite and infinite cyclic group?

Normally, a cyclic group is defined as a set of numbers generated by repeated use of an operator on a single element which is called the generator and is denoted by g.If the operation is multiplicative then the elements are g0, g1, g2, ...Such a group may be finite or infinite. If for some integer k, gk = g0 then the cyclic group is finite, of order k. If there is no such k, then it is infinite - and is isomorphic to Z(integers) with the operation being addition.


What is the symbol for integers?

Z is the symbol for integer. It is the initial letter of Zahlen, the German word meaning "number"