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.
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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.
no
The set of positive integers does not contain the additive inverses of all but the identity. It is, therefore, not a group.
Because the set is not closed under addition. If x and y are odd, then x + y is not odd.
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}.
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.
No. It is not a group.
Is the set of negative interferes a group under addition? Explain,
no
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.
The set of integers, under addition.
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.
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.
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}.
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.