Assignment 7
Properties of operations
Group number:______________________
Name:_______________________
Introduction.
Properties of whole-number operations make it easier to memorize basic facts and the
certain computations. If you learn 5+3=8, then you know what 3+5 equals. How would
you convince a first-grader that 3+5 is the same as 5+3?
Operations
An operation is a way to combine some elements of a set to get another element of the
set. You can think of an operation like a machine, with a certain number of slots for
inputs and a slot for the output. You put the inputs into the appropriate slots, turn a crank,
and out comes the output. You are already familiar with many operations: addition,
subtraction, multiplication and division are all operations. These four operations each
take two numbers as input, and give one number as output. They are called binary
operations.
1. Here are some examples of inputs and outputs for some binary operations. Identify
which operation is being performed inside the box.
a) Inputs: 3,2
Inputs: 1,6
Output: 5
Output: 7
b) Inputs: 11, 7
Inputs: 12, 12
Output: 77
Output: 144
2. Now see if you can work backward to figure out the algebraic rule for the operation ⊕
defined on the set {0,1,2,3} by the table at right.
0⊕0=______,
1⊕3=_____,
3⊕2=______.
⊕
0
1
2
3
0
1
1
1
1
1
1
2
3
4
2
1
3
5
7
3
1
4
7
10
Properties of Operations
3. Think about the four arithmetic operations on natural numbers. Write down in words
what the differences are among them, and keep these differences in mind as you read
through the following list of properties.
COMMUTATIVE: An operation is commutative if and only if the order of the two
things being operated on does NOT matter.
For example, the addition on the natural numbers is commutative because for any
whole numbers x and y, x+y=y+x
4. Try some examples and see whether you think the following operations on the natural
numbers are commutative. Give a counterexample for any operation that is not
commutative.
a) Subtraction (Is x – y = y – x?)
b) Multiplication (Is xy=yx?)
b) Division (Is x ÷ y = y ÷ x ? )
5. Is the operation ⊕ on {0,1,2,3} defined in question 2 commutative? Is there a visual
clue in the ⊕ table that let you know it is commutative or not?
ASSOCIATIVE: An operation is associative if and only if when you apply
the operation repeatedly, it does not matter where you put the parentheses.
For example, the addition on the natural numbers is associative, which is
illustrated by the fact that x+(y+z)=(x+y)+z.
6. Try some examples and see whether you think the following operations on the natural
numbers are associative. Give a counterexample for any operation that is not
commutative.
a) Subtraction: a-(b-c)=(a-b)-c?
b) Multiplication: a(bc)=(ab)c?
c) Division: a ÷ (b ÷ c) = (a ÷ b) ÷ c ?
IDENTITY: An operation has an identity element if one of the elements in the
set makes the operation leave all other elements alone.
For example, the addition on natural numbers has an identity element, namely 0,
because 0+a=a and a+0=a, no matter what a is.
7. Does the subtraction on natural numbers have an identity element? If so, what is the
element? (Can you fill the following blank: ____ - a = a - ____?)
8. Does the multiplication on natural numbers have an identity element? If so, what is the
element?
INVERSE: If an operation has an identity, then it may also have inverses.
The inverse of an element a is something which gets operated on along with a
in order to get the identity element.
For example, the addition on natural numbers has inverses, e.g., the inverse of 42
is –42, because adding 42 and –42 get you 0, the additive identity.
However, multiplication on the natural numbers does NOT have inverse, because
the multiplicative inverse of 2 is ½, which is not a natural number .Of course, if
an operation does not have an identity element, it can’t have inverses.
9. In all the following, the operation is defined on the set S={1,2,3,4}.
Try to explain whey each operation has (or does not have) the given properties.
Consider the operation @.
@
1
2
3
4
1
3
4
1
2
2
4
1
2
3
3
1
2
3
4
4
2
3
4
1
1) Is the operation @ commutative? Why? Give a few examples to support
your answer.
2) Is the operation @ associative? Why? Give a few examples to support
your answer.
3) What is the identity of @? Explain your answer.
4) The set S={1,2,3,4} does have inverses with respect to @. Why? Find the
inverse of each element.
Inv(1)=_____
Inv(2)=_____
Inv(3)=_____
Inv(4)=_____