Give the commutative property of multiplication and

A mathematical system consists of a set of elements and at least one _____ operation.

A binary operation is an operation or rule that can be performed on exactly two elements of a set, with the result being a(n) ______ element.

The commutative property of addition states: a + b = .

The commutative property of multiplication states: a # b = .

The associative property of multiplication states: (a # b) # c = .

The associative property of addition states: (a + b) + c = ___.

If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is under the given binary operation.

A specific example illustrating that a property is not true is called a(n) .

Since the sum of 0 and any integer is the given integer, we say that 0 is the additive element for the set of the integers under the operation of addition.

Since the product of 1 and any integer is the given integer, we say that 1 is the identity element for the set of the integers under the operation of multiplication.

When a binary operation is performed on two elements in a set and the result is the identity element for the binary operation, each element is said to be the of the other.

If a mathematical system possesses the following propertiesclosure, identity element, inverses, and the associative propertythen the mathematical system is a(n) .

A group that also satisfies the commutative property is called a(n) ______ (or abelian) group.

Another name for a commutative group is a(n) group.

Give the commutative property of multiplication and illustrate the property with an example.

Give the commutative property of addition and illustrate the property with an example.

Give the associative property of addition and illustrate the property with an example.

Give the associative property of multiplication and illustrate the property with an example.

Give an example to show that the commutative property does not hold for the set of integers under the operation of division.

Give an example to show that the commutative property does not hold for the set of integers under the operation of subtraction.

Give an example to show that the associative property does not hold for the set of integers under the operation of subtraction.

Give an example to show that the associative property does not hold for the set of integers under the operation of division.

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

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

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

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

Is the set of positive integers a group under the operation of subtraction?

Is the set of negative integers a group under the operation of division?

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

Is the set of integers a group under the operation of multiplication?

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

Is the set of negative integers a group under the operation of multiplication?

Is the set of rational numbers a group under the operation of addition?

Is the set of rational numbers a commutative group under the operation of multiplication?

Is the set of rational numbers a commutative group under the operation of division?

Is the set of rational numbers a group under the operation of subtraction?

Is the set of real numbers a group under the operation of addition?

Is the set of real numbers a group under the operation of multiplication?

Is the set of irrational numbers a group under the operation of addition?

Is the set of irrational numbers a group under the operation of multiplication?

Create a mathematical system with two binary operations. Select a set of elements and two binary operations so that one binary operation with the set of elements meets the requirements for a group and the other binary operation does not. Explain why the one binary operation with the set of elements is a group. For the other binary operation and the set of elements, find counterexamples to show that it is not a group.

There are other classifications of mathematical systems besides groups. For example, there are rings and fields. Do research to determine the requirements that must be met for a mathematical system to be (a) a ring and (b) a field. (c) Is the set of real numbers, under the operations of addition and multiplication, a field? Ask your instructor for references to use.