The Associative Property

The Associative Property:

A set has the associative property under a particular operation if the result of the operation is the same no matter how we group any sets of 3 or more elements joined by the operation. This definition will make more sense as we look at some examples.

More formally, if x, y and z are variables that represent any 3 arbitrary elements in the set we are looking at (let’s call the set we are looking at A), and the symbol # represents our operation, then the associative property for A with the operation # would be:

(x#y)#z = x#(y#z).

This means that the associative property only holds for the set A and the operation # if, no matter what elements we take from A and put in place of x, y and z, (x#y)#z will always give us the same result as x#(y#z).

Remember that the parentheses just tell us which pair to do first.

Notice that with all operations, the operation can only act on one pair of elements at a time. So the expression x#y#z doesn’t make sense, unless we set up a rule that tells us to go from left to right first or to go from right to left first, because there is no way for us to know which operation to do first: should we do x#y first, or y#z first? When we do addition or subtraction or multiplication or division, we have a rule that tells us just to go from left to right. But we don’t necessarily have this rule for arbitrary operations! If we are working with any operation that is not addition, subtraction, multiplication, or division, then we must have parentheses for any expression with more than two elements in it so that we know which occurrence of the operation to do first!

Letís look at some examples so that we can understand the associative property more clearly:

First let’s look at a few infinite sets with operations that are already familiar to us:

 

a) The set of natural numbers is associative under the operation of addition, because it is true that for any two natural numbers x, y, and z, (x+y)+z = x+(y+z).

 

b) The set of integers is not associative under the operation of division, because for any three integers x, y and z, there are many cases where (x ÷ y) ÷ z ≠ x ÷ (y ÷ z)! For example:

(10 ÷ 5) ÷ 2 = 2 ÷ 2 = 1

10 ÷ (5 ÷ 2) = 10 ÷ 2.5 = 2

1 ≠ 2!

So (10 ÷ 5) ÷ 2 ≠ 10 ÷ (5 ÷ 2)

So (x ÷ y) ÷ z = x ÷ (y ÷ z) is not true for all integers x, y and z!

to see some more examples!

c) The set of rational numbers is associative under the operation of multiplication, because it is true that for any three rational numbers x, y and z, (xy)z=x(yz). (Again, notice that we don’t always write out the operation symbol for multiplication. It is traditional when we write multiplication to leave the multiplication symbol out so that x×y is just written xy.)

 

d) The set of real numbers is not associative under the operation of subtraction, because for any three real numbers x, y and z, there are many cases where (x – y) – z ≠ x – (y – z)! For example:

(2 – 5) – 12 = (–3) – 12 = –15

2 – (5 – 12) = 2 – (–7) = 2 + 7 = 9

–15 ≠ 9!

So (2 – 5) – 12 ≠ 2 – (5 – 12)

So (x – y) – z = x – (y – z) is not true for all real numbers x, y and z!

It can be a little bit more difficult, if all we have is an operation table, to tell whether or not a set has the associative property under a particular operation. To figure out how to do this,

Letís look at a few simple sets with operation tables and check to see if they have the associative property.†

e) Here is an operation table for the set {a,b,c} and the operation *:

*

a

b

c

a

a

b

c

b

b

a

c

c

c

c

a

To check associativity, we must check every possible instance of the equation (x*y)*z = x*(y*z). That means we must think of every possible combination of what x, y, and z could be.

 

Using the above table to write out every possible combination of elements that could go in for x, y, and z and then calculating (x*y)*z and comparing it to x*(y*z) gives us the following table:

 

Values of x, y, z
(x*y)*z
x*(y*z)
Does (x*y)*z = x*(y*z)?
x=a, y=a, z=a
(a*a)*a = a*a = a
a*(a*a) = a*a = a
a = a, So (a*a)*a = a*(a*a).
x=a, y=a, z=b
(a*a)*b = a*b = b
a*(a*b) = a*b = b
b = b, So (a*a)*b = a*(a*b).
x=a, y=a, z=c
(a*a)*c = a*c = c
a*(a*c) = a*a = c
c = c, So (a*a)*c = a*(a*c).
x=a, y=b, z=a
(a*b)*a = b*a = b
a*(b*a) = a*b = b
b = b, So (a*b)*a = a*(b*a).
x=a, y=b, z=b
(a*b)*b = b*b = a
a*(b*b) = a*a = a
a = a, So (a*b)*b = a*(b*b).
x=a, y=b, z=c
(a*b)*c = b*c = c
a*(b*c) = a*c = c
c = c, So (a*b)*c = a*(b*c).
x=a, y=c, z=a
(a*c)*a = c*a = c
a*(c*a) = a*c = c
c = c, So (a*c)*a = a*(c*a).
x=a, y=c, z=b
(a*c)*b = c*b = c
a*(c*b) = a*c = c
c = c, So (a*c)*b = a*(c*b).
x=a, y=c, z=c
(a*c)*c = c*c = a
a*(c*c) = a*a= a
a = a, So (a*c)*c = a*(c*c).
x=b, y=a, z=a
(b*a)*a = b*a = b
b*(a*a) = b*a = b
b = b, So (b*a)*a = b*(a*a).
x=b, y=a, z=b
(b*a)*b = b*b = a
b*(a*b) = b*b = a
a = a, So (b*a)*b = b*(a*b).
x=b, y=a, z=c
(b*a)*c = b*c = c
b*(a*c) = b*c = c
c = c, So (b*a)*c = b*(a*c).
x=b, y=b, z=a
(b*b)*a = a*a = a
b*(b*a) = b*b = a
a = a, So (b*b)*a = b*(b*a).
x=b, y=b, z=b
(b*b)*b = a*b = b
b*(b*b) = b*a = b
b = b, So (b*b)*b = b*(b*b).
x=b, y=b, z=c
(b*b)*c = a*c = c
b*(b*c) = b*c = c
c = c, So (b*b)*c = b*(b*c).
x=b, y=c, z=a
(b*c)*a = c*a = c
b*(c*a) = b*c = c
c = c, So (b*c)*a = b*(c*a).
x=b, y=c, z=b
(b*c)*b = c*b = c
b*(c*b) = b*c = c
c = c, So (b*c)*b = b*(c*b).
x=b, y=c, z=c
(b*c)*c = c*c = a
b*(c*c) = b*a = b
a ≠ b, So (b*c)*c ≠ b*(c*c)!
x=c, y=a, z=a
(c*a)*a = c*a = c
c*(a*a) = c*a = c
c = c, So (c*a)*a = c*(a*a).
x=c, y=a, z=b
(c*a)*b = c*b = c
c*(a*b) = c*b = c
c = c, So (c*a)*b = c*(a*b).
x=c, y=a, z=c
(c*a)*c = c*c = a
c*(a*c) = c*c = a
a = a, So (c*a)*c = c*(a*c).
x=c, y=b, z=a
(c*b)*a = c*a = c
c*(b*a) = c*b = c
c = c, So (c*b)*a = c*(b*a).
x=c, y=b, z=b
(c*b)*b = c*b = c
c*(b*b) = c*a = c
c = c, So (c*b)*b = c*(b*b).
x=c, y=b, z=c
(c*b)*c = c*c = a
c*(b*c) = c*c = a
a = a, So (c*b)*c = c*(b*c).
x=c, y=c, z=a
(c*c)*a = a*a = a
c*(c*a) = c*c = a
a = a, So (c*c)*a = c*(c*a).
x=c, y=c, z=b
(c*c)*b = a*b = b
c*(c*b) = c*c = a
b ≠ a, So (c*c)*b ≠ c*(c*b)!
x=c, y=c, z=c
(c*c)*c = a*c = c
c*(c*c) = c*a = c
c = c, So (c*c)*c = c*(c*c).

 

So the set {a,b,c} under the operation * is not associative because we have at least one example where (x*y)*z ≠ x*(y*z)! In fact we have two examples when this fails: (b*c)*c ≠ b*(c*c) and (c*c)*b ≠ c*(c*b)! When x is b and y is c and z is c, (x*y)*z ≠ x*(y*z)! And when x is c and y is c and z is b (x*y)*z ≠ x*(y*z)!

 

 

While there is really no good shortcut you can use to try to see if a set under an operation given by an operation table is associative, there are a few things you can look for which may cut down the number of instances you need to check:

If a set has an identity, then we don't need to check any instances which include the identity. Why? Let's take a look at what happens if we compute an instance which contains the identity. For this purpose, we will let e represent the identity element of our set, and we will let x and y represent any two elements of our set.

Let's see if (e*x)*y= e*(x*y) is true.

First we will compute (e*x)*y. Since the identity by definition does not change elements when it acts on them with the operation, we must have e*x=x. So (e*x)*y=x*y.

Now we will compute e*(x*y). Again, since the identity by definition does not change elements when it acts on them with the operation, we must have e*(x*y)=x*y. So (e*x)*y= e*(x*y) is true.

We can use similar reasoning to see that (x*e)*y= x*(e*y) is true and that (x*y)*e= x*(y*e) is true.

So expressions which contain the identity will always be associative. This means that in order to check whether or not a set has the associative property, we need only check expressions which contain only elements which are NOT the identity.

This can cut down considerably on the number of items we check. To see this, let's look at our example again:

Example:

Here is an operation table for the set {a,b,c} and the operation *:

*

a

b

c

a

a

b

c

b

b

a

c

c

c

c

a

Notice that this set has an identity: a, because whenever a is multiplied by any element in the set on either the right or the left, it leaves that element unchanged.

So, to check associativity, we must check every possible instance of the equation (x*y)*z = x*(y*z) which does not include a (becuase a is the identity, and we don't need to check combinations which include the identity).

 

Values of x, y, z
(x*y)*z
x*(y*z)
Does (x*y)*z = x*(y*z)?
x=b, y=b, z=b
(b*b)*b = a*b = b
b*(b*b) = b*a = b
b = b, So (b*b)*b = b*(b*b).
x=b, y=b, z=c
(b*b)*c = a*c = c
b*(b*c) = b*c = c
c = c, So (b*b)*c = b*(b*c).
x=b, y=c, z=b
(b*c)*b = c*b = c
b*(c*b) = b*c = c
c = c, So (b*c)*b = b*(c*b).
x=b, y=c, z=c
(b*c)*c = c*c = a
b*(c*c) = b*a = b
a ≠ b, So (b*c)*c ≠ b*(c*c)!
x=c, y=b, z=b
(c*b)*b = c*b = c
c*(b*b) = c*a = c
c = c, So (c*b)*b = c*(b*b).
x=c, y=b, z=c
(c*b)*c = c*c = a
c*(b*c) = c*c = a
a = a, So (c*b)*c = c*(b*c).
x=c, y=c, z=b
(c*c)*b = a*b = b
c*(c*b) = c*c = a
b ≠ a, So (c*c)*b ≠ c*(c*b)!
x=c, y=c, z=c
(c*c)*c = a*c = c
c*(c*c) = c*a = c
c = c, So (c*c)*c = c*(c*c).

Notice that by only checking the instances which do not contain the identity a, we have narrowed down the number of combinations we need to check from 27 to 8! This is a big improvement!

Be careful! Many students make the mistake of concluding that a set is associative by checking just a few examples. You CANNOT do this! To determine whether or not a set is associative, YOU MUST CHECK EVERY SINGLE COMBINATION OF 3 ELEMENTS, unless you have a good general argument for why all combinations will be associative or you have a good reason (such as the existence of an identity element) for limiting the number of cases you must check. (Notice that even when you have the existence of an identity element, you still have to check ALL the cases which do not include the identity element.)

 

Now return to Blackboard to answer Group Lecture Questions 6: Associativity!