Math and Arithmetic
Algebra

# 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

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.

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## Related Questions

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.

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

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

Yes, with respect to multiplication but not with respect to addition.

Because the set is not closed under addition. If x and y are odd, then x + y is not odd.

A cyclic group, by definition, has only one generator. An example of an infinite cyclic group is the integers with addition. This group is generated by 1.

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.

It defines 0 as the identity in the group of numbers with respect to addition.

Say we have a group G, and some subgroup H. The number of cosets of H in G is called the index of H in G. This is written [G:H].If G and H are finite, [G:H] is just |G|/|H|.What if they are infinite? Here is an example. Let G be the integers under addition. Let H be the even integers under addition, a subgroup. The cosets of H in G are H and H+1. H+1 is the set of all even integers + 1, so the set of all odd integers. Here we have partitioned the integers into two cosets, even and odd integers. So [G:H] is 2.

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}.

negative is - Positive is + The group of integers is represented by &#8484;.

In abstract algebra, a group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory. Many of the structures investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Other important examples are the group of non-singular matrices under multiplication and the group of invertible functions under composition. Group theory allows for the properties of such structures to be investigated in a general setting. Group theory has extensive applications in mathematics, science, and engineering. Many algebraic structures such as fields and vector spaces may be defined concisely in terms of groups, and group theory provides an important tool for studying symmetry, since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields.

Rationals and Reals. The answer also belongs to Natural numbers and Integers, but these sets are not closed with respect to division and so are not groups.

It follows from the fact that real numbers area group with respect to addition and the decimal representation of numbers.

The rational numbers form an algebraic structure with respect to addition and this structure is called a group. And it is the property of a group that every element in it has an additive inverse.

What you need to know about integers is that integers is the name for the group of numbers that include whole numbers and negative numbers. But integers DO NOT include fractions.

0, zero, is defined as the identity element for addition and subtraction. * * * * * While 0 is certainly the identity element with respect to addition, there is no identity element for subtraction. The identity element of a set, for a given operation, must commute with every element of the set. Since a - 0 &ne; 0 - a, according to group theory, 0 is not an identity with respect to subtraction.

No. One of the group axioms is that each element must have an inverse element. This is not the case with integers. In other words, you can't solve an equation like: 5 times "n" = 1 in the set of integers.

To be a group, the set of integers with multiplication has to satisfy certain axioms: - Associativity: for all integers x,y and z: x(yz) = (xy)z - Identity element: there exists some integer e such that for all integers x: ex=xe=x - Inverse elements: for every integer x, there exists an integer y such that xy=yx=e, where e is the identity element The associativity is satisfied and 1 is clearly the identity element, however no integer other than 1 has an inverse as in the integers xy = 1 implies x=y=1

The discovery of the noble gases led to the addition of the group 0, which is also designated as group 18/VIIIA.

Positive integers are greater than zero, negative integers are less than zero. The set of positive integers is closed under multiplication (and form a group), the set of negative integers is not.

They are both whole numbers (integers) and natural numbers.All natural numbers are integers, but integers is a larger group of numbers.The group consists of the natural numbers, zero and the whole negative numbers (e.g. '-4' and '-560').

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