MATH CH 9

A mathematical system is a commutative group if all five of the following conditions hold.

1. The set of elements is closed under the given operation. 2. An identity element exists for the set under the given operation. 3. Every element in the set has an inverse under the given operation. 4. The set of elements is associative under the given operation. 5. The set of elements is commutative under the given operation.

(a+b)+c=a+(b+c)

Associative Property of Addition

identity element

a transformation that leaves an object unchanged

the commutative property of addition.

a+ b = b + a​, for any elements a and b Ex: 3+4=4+3

Since clock 12 arithmetic under the operation of addition contains only the elements​ {1,2,3,4,5,6,7,8,9,10,11,12}, the mathematical system is

closed

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

closed under the given binary operation.

In clock 12 arithmetic under the operation of​ addition, since the system is​ closed, there is an identity​ element, each element has an​ inverse, and the associative and commutative properties​ hold, the mathematical system is a ___________________ group.

commutative

Every modulo system under the operation of addition meets all five requirements needed for it to be called a

commutative group

If a and b have the same remainder when divided by​ m, then a is_________ to b modulo m.

congruent

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.

identity

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

inverse

an _______ element for A is an element which gives the identity when composed with A

inverse element

In clock 12​ arithmetic, since 1+11= 11+1 =12, we say that 1 and 11 are additive

inverses

the commutative property of addition tells us

numbers can be added in any order and you will still get the same answer. The formula for this property is a + b = b + a.

when decrypting code,

subtract

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

For any elements​ a, b, and​ c, ​(a+​b)+c=a+​(b+c). An example of this is ​(3+​2)+6=3+(2+​6).

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

No, it is not a commutative group. There is no identity element in the set of negative integers under the operation of addition. No, it is not a commutative group. There is at least one negative integer that does not have an inverse in the set of negative integers under the operation of addition.

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

No, it is not a group. There is at least one positive integer that does not have an inverse in the set of positive integers under the operation of addition. No, it is not a group. There is no identity element in the set of positive integers under the operation of addition.

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

Yes, it is a group.

commutative group

a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

The associative property states that

you can add or multiply regardless of how the numbers are grouped. By 'grouped' we mean 'how you use parenthesis'. In other words, if you are adding or multiplying it does not matter where you put the parenthesis. Add some parenthesis any where you like!.

is the set of integers a commutative group under the operation of​ addition?

​Yes; it satisfies the five properties needed.

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

​​No, it is not a commutative group. There exist rational numbers​ a, b, and c such that ​(a/b) /c not equal to a/​(b/ ​c). No, it is not a commutative group. The set of rational numbers is not closed under the operation of division No, it is not a commutative group. There exist rational numbers a and b such that a/b not equal to b/a. No, it is not a commutative group. There is at least one rational number that does not have an inverse in the set of rational numbers under the operation of division.