Quantitative Skills Test 2

Associative Property of Addition

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

Subtraction Identity

0

Multiplicative identity

1

5 properties of a commutative group

1. The set of elements is closed under a given operation. 2. All identity elements exist for the set under the operation. 3. Every element has an inverse 4. Set of elemets associative

Clock 12 arithmetic under the operation of addition only contains elements 1,2,3,4,5,6,7,8,9,10,11,12 making the mathematical system

Closed

whole numbers

Natural numbers ( counting numbers) and zero; 0, 1, 2, 3....

Is the set of positive integers a group 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. There is also at least one positive integer that does not have an inverse in the set of positive integers under the operation of addition.

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

No, it's 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. There is also no identity element in the set of negative integers under the operation of addition.

closed set

Result has to be in that set after a binary option is performed on any elements in the set.

Integers

The set of whole numbers and their opposites.

In clock 12 arithmetic, since 1+11=11+1=12

We say 1 and 11 are additive inverses.

Inverse

When an element operates in its inverse the result is the identity element.

identity

When binary operation performed the result is the given element. Ex: whole numbers/ addition Additive identity= 0

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

Yes

Commutative Property of Addition

a + b = b + a

rational numbers

any number that can be written as a fraction.

Modular Arithmetic

examining only the remainders of numbers. Example: in mod 12 the 12 is represented as a 0.