5.1: Operations and Properties

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This exercise set is designed to give you an understanding of what "binary operations" are, and to give you a deeper understanding for the commutative, associative and distributive properties. To do this, we're going to define and work with some nonsense operations. First, we need to get a good grasp on what an operation is. The binary operations you are familiar with are addition, subtraction, multiplication and division. This means that you are performing a rule using two numbers. For instance, we know what to do when we see the plus sign (+), the subtraction sign (–), the multiplication sign ($\times$ or $\bullet$ ) or the division sign ($\div$) between two numbers. There is a specific "rule" that we apply.

Suppose you were asked to compute 5 )( 3. You wouldn't know what to do unless someone told you what )( meant. It's like asking someone who has never heard of addition, or seen an addition sign (+) to compute 5 + 3. In order to do a computation, the operation used must be defined. The operations you already know about are addition, multiplication, subtraction and division. You also know how to compute with exponents, and how to compare numbers (<, = or >).

Let's start off by defining what )( means. This is just a made-up operation that we're defining to be used in this workbook, so that you'll have a deeper understanding of binary operations. There really isn't such an operation as i in the real world.

Define )( like this: M )( N = 3M + 2N + 8.

The variables I use to define this binary operation are arbitrary. I could use any letters or symbols I want. Or I could explain how to perform the binary operation )( , which is much more cumbersome.

This is how I could explain how to compute: To compute with )( , multiply the number before the )( by 3 and add this to twice the number after the )( , and then add 8. As you can see, it is easier to "explain" by using variables. Here are three more ways I could have defined )( , without changing the meaning of )(.

a )( b = 3a + 2b + 8

b )( a = 3b + 2a + 8

x )( y = 3x + 2y + 8

Notice that in each case, one multiplies the first symbol by 3, adds it to twice the second symbol, and then adds 8! It's the same rule, but I chose different variables to "explain" the rule. You have to pay careful attention to what the meaning of the operation is, and follow the rule exactly. It's like working with functions in algebra.

Let's compute and simplify a few problems with )( :

5 )( 3 = 3(5) + 2(3) + 8 = 15 + 6 + 8 = 29

3 )( 5 = 3(3) + 2(5) + 8 = 9 + 10 + 8 = 27

r )( s = 3r + 2s + 8

The rule for the operation doesn't necessarily depend on both of the variables used, and in fact, may not depend on either of them. On the next page, several new operations are defined; some computations use only one of the variables, some use neither. Suppose there are eight new binary operations that are defined as follows:

)(	a )( b = 3a + 2b + 8
*	a * b = a + 2b
,	a, b = $a^{2} + b^{2}$
!	a ! b = 2 (Notice the answer doesn't depend on a or b)
$\oplus$	$a \oplus b$ = 3ab
#	a # b = the smaller value of a or b
@	a @ b = 2b (Notice the answer doesn't depend on a)
$\odot$	a $\odot$ b = 2a + 2b
$\boxed{\times}$	a $\boxed{\times}$ $b = a^{2} + b$

Remember that a and b are just "dummy" variables. Any variables could have been used to define the above functions. The first operation, *, could have been defined like this: m * n = m + 2n. The meaning of the definition is exactly the same. To apply the definition of * to get the answer, it says to take the first number and add it to twice the second number.

Here are a several examples for you to study before going on to the next page

6 )( 7 = 3(6)+2(7)+8 = 18+14+8 = 40	2 $\oplus$ 5 = 3(2)(5) = 30
7 )( 6 = 3(7)+2(6)+8 = 21+12+8= 41	r $\oplus$ s = 3rs
4 )( 4 = 3(4)+2(4)+8 = 12+8+8 = 28	4 # 7 = 4
v )( z = 3v + 2z + 8	5 # 2 = 2
5 * 3 = 5 + 2(3) = 5 + 6 = 11	7 # 4 = 4
4 * 7 = 4 + 2(7) = 4 + 14 = 18	c # d = the smaller value of c or d
3 * 5 = 3 + 2(5) = 3 + 10 = 13	5 @ 6 = 2(6) = 12
v * z = z + 2z	7 @ 3 = 2(3) = 6
5 , 3 = $5^{2} + 3^{2} = 25 + 9 = 34$	6 @ 5 = 2(5) = 10
4 , 3 = $4^{2} + 3^{2}$ = 16 + 9 = 25	p @ q = 2q
3 , 5 = $3^{2} + 5^{2}$ = 9 + 25 = 34	6 $\odot$ 3 = 2(6) + 2(3) = 12 + 6 = 18
v , z = $v^{2} + z^{2}$	5 $\odot$ 8 = 2(5) + 2(8) = 10 + 16 = 26
5 ! 3 = 2	3 $\odot$ 6 = 2(3) + 2(6) = 6 + 12 = 18
8 ! 7 = 2	h $\odot$ k = 2h + 2k
(junk) ! (stuff) = 2	4 $\boxed{\times}$ $7 = 4^{2}$ + 7 = 16 + 7 = 23
w ! q = 2	6 $\boxed{\times}$ $9 = 6^{2}$ + 9 = 36 + 9 = 45
$5 \oplus 2 = 3(5)(2) = 30$	7 $\boxed{\times}$ $4 = 7^{2}$ + 4 = 49 + 4 = 53
$4 \oplus 7 = 3(4)(7) = 84$	z $\boxed{\times}$ $n = z^{2} + n$

Exercise 1

Compute the following. Show all of the steps. If you need help, look at the examples on the previous page

7 )( 6 = ______________________	4 $\oplus$ 7 = _____________________
4 )( 4 = ______________________	2 $\oplus$ 5 = _____________________
v )( z = ______________________	r $\oplus$ s = _____________________
5 * 3 = _____________________	4 # 7 = _____________________
4 * 7 = _____________________	5 # 2 = _____________________
3 * 5 = _____________________	7 # 4 = _____________________
v * z = _____________________	c # d = _____________________
5 , 3 = _____________________	5 @ 6 = ____________________
4 , 3 = _____________________	7 @ 3 = ____________________
3 , 5 = _____________________	6 @ 5 = ____________________
v , z = ______________________	p @ q = ____________________
5 ! 3 = _____________________	6 $\odot$ 3 = _____________________
8 ! 7 = _____________________	5 $\odot$ 8 = ____________________
(junk) ! (stuff) = _____________	3 $\odot$ 6 = ____________________
w ! q = ____________________	h $\odot$ k = ____________________
$5 \oplus 2$ = _____________________	4 $\boxed{\times}$ 7 = ____________________
	6 $\boxed{\times}$ 9 = ____________________
	7 $\boxed{\times}$ 4 = ____________________
	z $\boxed{\times}$ n = ____________________

The above problems are the same examples that were done on the previous page. If you need help, look back at the examples again. Don't go on until you can get them all right.

Exercise 2

Compute and simplify the following. Show all of the steps.

a. 8 * 4

b. 4 , 7

c. 79 ! 88

d. 7 $\oplus$ 2

e. 6 # 4

f. 4 @ 9

g. 5 $\odot$ 2

h. 6 $\boxed{\times}$ 5

i. 2 )( 3

An operation, $\blacklozenge$, is commutative if for any two values, X and Y, X $\blacklozenge$ Y = Y $\blacklozenge$ X.

Again, $\blacklozenge$ is just a "dummy" operation and "X" and "Y" are dummy variables. For a particular operation to be commutative, the equation must always be true no matter what values are used for X and Y.

For instance, the operation * is commutative only if m * n = n * m is always true no matter what values are put in for m or n.

To show that an operation is not commutative, all you need to do is provide a counterexample (with particular values) that shows the equation is not true for at least those particular values. To prove an operation is commutative is more involved because you must prove it is always true no matter what values you use. You would have to switch the order of the original values (a and b, or X and Y, etc.), and show algebraically that both expressions simplify to the same thing.

From my examples after defining the operations and the problems you worked in exercise 2, it should be clear which of the eight operations are not commutative.

Let @ be defined as follows: m @ n = 2n. Is @ commutative?

Solution: If @ is commutative, then m @ n = n @ m for all values m and n.

But, 5 @ 6 = 12 and 6 @ 5 = 10.

Therefore, @ is not commutative since 5 @ 6 $\neq$ 6 @ 5.

Let & be defined as follows: m & n = 2mn. Is & commutative?

Solution: If & is commutative, then m & n = n & m for all values m and n. First, I'd try some numbers in for a and b to see if I might come up with a counterexample. For instance, 5 & 6 = 2(5)(6) = 60 and 6 & 5 = 2(6)(5) = 60. No counterexample here. So, I use algebra to prove that m & n = n & m. Since m & n = 2mn, and n & m = 2nm, the question is: Does 2mn = 2nm? Yes, it does! Since m & n = n & m for all m and n, then & is commutative.

Exercise 3

For each operation listed, determine whether it is commutative or not. If it is not commutative, give a counterexample, like I did for @. If it is commutative, prove it is commutative, like I did for & above. Begin each problem by stating what equation must be true if the operation listed is commutative

a. ! is defined like this: m ! n = 2. Determine if ! is commutative.

Write the general equation that is true if ! is commutative: _____________

Is ! commutative? ________. If you answered yes, prove ! is commutative. If you answered no, provide a counterexample to illustrate it is not commutative.

b. $\oplus$ is defined like this: m $\oplus$ n = 3mn. Determine if $\oplus$ is commutative.

Write the general equation that is true if $\oplus$ is commutative: ______________

Is $\oplus$ commutative? ________. If you answered yes, prove $\oplus$ is commutative. If you answered no, provide a counterexample to illustrate it is not commutative.

c. # is defined: m # n = the smaller value of m or n. Determine if # is commutative.

Write the general equation that is true if # is commutative: ______________

Is # commutative? ________. If you answered yes, provide an example. If you answered no, provide a counterexample to illustrate it is not commutative

d. $\odot$ is defined like this: m $\odot$ n = 2m + 2n. Determine if $\odot$ is commutative.

Write the general equation that is true if $\odot$ is commutative:

Is $\odot$ commutative? ________. If you answered yes, prove $\odot$ is commutative. If you answered no, provide a counterexample to illustrate it is not commutative.

e. $\boxed{\times}$ is defined like this: m $\boxed{\times}$ $n = m^{2} + n$. Determine if $\boxed{\times}$ is commutative.

Write the general equation that is true if $\boxed{\times}$ is commutative:

Is $\boxed{\times}$ commutative? ________. If you answered yes, prove $\boxed{\times}$ is commutative. If you answered no, provide a counterexample to illustrate it is not commutative.

f. $m , n = m^{2} + n^{2}$. Determine if o is commutative.

Write the general equation that is true if , is commutative:

Is , commutative? ________. If you answered yes, prove , is commutative. If you answered no, provide a counterexample to illustrate it is not commutative.

g. m * n = m + 2n. Determine if * is commutative.

Write the equation that must be true if * is commutative:

Is * commutative? ________. If you answered yes, prove * is commutative. If you answered no, provide a counterexample to illustrate it is not commutative.

h. m )( n = 3m + 2n + 8 . Determine if )( is commutative.

Write the general equation that is true if )( is commutative:

Is )( commutative? ________. If you answered yes, prove )( is commutative. If you answered no, provide a counterexample to illustrate it is not commutative.

Before going on to determining whether an operation is associative or distributive, we should compute a few more problems, which are a bit more involved. Make sure you follow the order of operations as you work through these next few problems. Look at the examples first

Example 1:

$(2 \oplus 3) \oplus 4$

(first do $2 \oplus 3) \quad 18 \oplus 4$

216

Example 2:

4, (3, 2)

(first do 3 , 2) 4, 13

185

Exercise 4

Simplify each of the following. Do the order of operations (do what is in parentheses first) and show each step.

a. (3 * 5) * 2	b. (3 @ 5) @ 2	c. (3 ! 5) ! 7
d. (3 $\oplus$ 4) $\oplus$ 2	e. (3 $\odot$ 5) $\odot$ 2	f. (3 $\boxed{\times}$ 2) $\boxed{\times}$ 4

Exercise 5

Compute the following, using the definitions for the operations as shown above. Note that more than one operation is in some of the problems. When simplifying, use the order of operations (do what is in parentheses first) and show each step.

a. 3 $\oplus$ (5 $\oplus$ 2)	b. 3 $\odot$ (5 $\odot$ 2)	c. 3 $\boxed{\times}$ (4 $\boxed{\times}$ 2)
d. 8 # (9 # 6)	e. (8 # 9) # 6	f. (4 , 3) , 2
g. 2 @ (3 # 4)	h. (2 @ 3) # 4	i. (1 $\odot$ 5) # 40

An operation, $\blacklozenge$, is associative if (X $\blacklozenge$ Y) $\blacklozenge$ Z = X $\blacklozenge$ (Y $\blacklozenge$ Z) for values of X, Y and Z.

Again, $\blacklozenge$ is just a "dummy" operation and "X" and "Y" and "Z" are dummy variables. For a particular operation to be associative, the equation must always be true no matter what values are used for X, Y and Z.

For instance, the operation * is associative only if (v * w) * x = v * (w * x) is always true no matter what values are put in for v, w or x.

To show that an operation is not associative, all you need to do is provide a counterexample (using actual numbers) that shows the equation is not true for at least those particular numbers. To prove an operation is associative is more involved because you must prove it is always true no matter what values you use. You would have to switch the parentheses, and show algebraically that both expressions always simplify to the same thing.

Example

Let @ be defined as follows: m @ n = 2n. Is @ associative?

If @ is associative, then (a @ b) @ c = a @ (b @ c) for all values a,b and c. First, I'd try some numbers in for a, b and c to see if I might come up with a counterexample: (2 @ 3) @ 4 = 6 @ 4 = 8, and 2 @ (3 @ 4) = 2 @ 8 = 16. This shows @ is not associative and provides us with a counterexample: Since $(2 @ 3) @ 4 \neq 2 @ (3 @ 4)$, @ is not associative.

Example

Let & be defined as follows: m & n = 2mn. Is & associative?

If & is associative, then (a & b) & c = a & (b & c) for all values a,b and c. First, I'd try some numbers in for a, b and c to see if I might come up with a counterexample: (2 & 3) & 4 = 12 & 4 = 96, and 2 & (3 & 4) = 2 & 24 = 96. No counterexample here. I might want to try another example with numbers, or I can go directly to using algebra to see if I can prove that it is always true that (a & b) & c = a & (b & c). First, we need to simplify the left side: (a & b) & c = 2ab & c = 4abc. Now, we have to simplify the right side: a &(b & c)=a &2bc = 4abc. Since (a &b)& c = a & (b & c), then & is associative.

Exercise 6

For each operation listed, determine whether it is associative or not. If it is not associative, give a counterexample, like I did for @ and )(. If it is associative, prove it is associative, like I did for & above. Begin each problem by stating the general equation that is true if the operation listed is associative.

a. ! is defined like this: m ! n = 2. Determine if q is associative.

Write the general equation that is true if ! is associative:

Is ! associative? ________. If you answered yes, prove ! is associative. If you answered no, provide a counterexample to illustrate it is not associative

b. $\oplus$ is defined like this: m $\oplus$ n = 3mn. Determine if $\oplus$ is associative.

Write the general equation that is true if $\oplus$ is associative:

Is $\oplus$ associative? ________. If you answered yes, prove $\oplus$ is associative. If you answered no, provide a counterexample to illustrate it is not associative.

c. m # n = the smaller value of m or n. Determine if # is associative.

Write the general equation that is true if # is associative:

Is # associative? ________. If you answered yes, provide an example. If you answered no, provide a counterexample to illustrate it is not associative.

d. $\odot$ is defined like this: m $\odot$ n = 2m + 2n. Determine if $\odot$ is associative.

Write the general equation that is true if $\odot$ is associative:

Is $\odot$ associative? ________. If you answered yes, prove $\odot$ is associative. If you answered no, provide a counterexample to illustrate it is not associative.

e. $\boxed{\times}$ is defined like this: m $\boxed{\times}$ $n = m^{2} + n$. Determine if $\boxed{\times}$ is associative.

Write the general equation that is true if $\boxed{\times}$ is associative:

Is $\boxed{\times}$ associative? ________. If you answered yes, prove $\boxed{\times}$ is associative. If you answered no, provide a counterexample to illustrate it is not associative.

f. m, $n = m^{2} + n^{2}$. Determine if , is associative.

Write the general equation that is true if , is associative:

Is , associative? ________. If you answered yes, prove , is associative. If you answered no, provide a counterexample to illustrate it is not associative.

g. a * b = a + 2b. Determine if * is associative.

Write the general equation that is true if * is associative:

Is * associative? ________. If you answered yes, prove * is associative. If you answered no, provide a counterexample to illustrate it is not associative.

An operation, $\blacklozenge$, distributes over another operation, $\phi$ if for any values of X, Y and Z:

X $\blacklozenge$ (Y $\phi$ Z) = (X $\blacklozenge$ Y) $\phi$ (X $\blacklozenge$ Z). This is a Left-Hand Distributive Property, because the symbol on the LEFT (in this case an X) is being distributed across the parentheses to the right. The Right-Hand Distributive Property states: An operation, $\blacklozenge$, distributes over

another operation, $\phi$ if for any values of X, Y and Z: (Y $\phi$ Z) $\blacklozenge$ X = (Y $\blacklozenge$ X) $\phi$ (Z $\blacklozenge$ X).

Unless otherwise stated, assume that the distributive property refers to the Left-Hand Distributive Property.

Remember that $\blacklozenge$ and $\phi$ are just "dummy" operations and "X" and "Y" and "Z" are dummy variables. For a particular operation to distributive over another operation, the equation

must always be true no matter what values or variables are used for X, Y and Z.

For instance, the operation * distributes over + only if v * (w + x) = (v * w) + (v * x) is always true no matter what value you put in for v, w and x.

To show that an operation does not distribute over another operation, you only need to provide a counterexample (using actual numbers) that shows the equation is not true for at least those particular numbers. To prove an operation does distribute over another operation is more involved because you must prove it is always true no matter what values you use. You would first have to work the left side of the equation (by using order of operations — simplifying in parentheses first), and then work the right side of the equation (by using order of operations by simplifying in parentheses first), and finally you would need to show algebraically that both expressions always simplify to the same thing.

Exercise 7

Let @ be defined as follows: m @ n = 2n. We're going to determine if @ distributes over addition. Write the equation that would be true if @ distributed over addition:

I'll help you with the rest of the solution. If @ distributes over addition, then a @ (b + c) = (a @ b) + (a @ c) for all values a, b and c. First, I'd try some numbers in for a, b and c to see if I might come up with a counterexample: 5 @ (3 + 4) = 5 @ 7 = 14 and (5 @ 3) + (5 @ 4) = 6 + 8 = 14. No counterexample here. I can try another example with numbers or try proving it algebraically. First, simplify the left side using the definition of @: a @ (b + c) = 2(b + c) = 2b + 2c. Now to simplify the right side: (a @ b) + (a @ c) = 2b + 2c. Since both expressions equal the same thing (2a + 2b), a @ (b + c) = (a @ b) + (a @ c), and therefore, we say that YES, @ distributes over addition.

Exercise 8

Let @ be defined as follows: m @ n = 2n. We are going to determine if addition distributes over @. Write the equation that is true if addition distributes over @:

I'll help you with the rest of the solution. If addition distributes over @, then a + (b @ c) = (a + b) @ (a + c) for all values a,b and c. First, I'd try some numbers in for a, b and c to see if I might come up with a counterexample: 5 + (3 @ 4) = 5 + 8 = 13 and (5 + 3) @ (5 + 4) = 8 @ 9 = 18. This shows that addition does not distribute over @ and provides us with a counterexample.
Since $5 + (3 @ 4) \neq (5 + 3) @ (5 + 4)$, addition does not distribute over @.

Exercise 9

Let & be defined by: m & n = 2mn and let $ be defined by: m $ $n = m^{2}$. We are going to determine if & distributes over $ or if $ distributes over &. First, let's determine if & distributes over $.

a. Write the equation that would be true if & distributed over $:

I'll help you with the rest of the solution. If & distributes over $, then this equation is true: a & (b $ c) = (a & b) $ (a & c). Let's compute each side of the equation by putting in some values for a, b and c to see if we find a counterexample. We'll see if 2 & (3 $ 4) and (2 & 3) $ (2 & 4) are equal. Since 2 & (3 $ 4) = 2 & 9 = 36, and
(2 & 3) $ (2 & 4) = 12 $ 16 = 144, the equation isn't true and we have a counterexample. Therefore, & does not distribute over $.

Next, we'll determine if $ distributes over &.

b. Write the equation that is true if $ distributes over &:

I'll help you with the rest of the solution. If $ distributes over &, then this equation is true for all values of a, b and c: a $ (b & c) = (a $ b) & (a $ c). Let's compute each side of the equation by putting in some values for a, b and c to see if we find a counterexample. Let's see if 2 $ (3 & 4) and (2 $ 3) & (2 $ 4) are equal. Since 2 $ (3 & 4) = 2 $ 24 = 4, and (2 $ 3) & (2 $ 4) = 4 & 4 = 32, the equation isn't true (since $4 \neq 32$) and we have a counterexample. Therefore, $ does not distribute over &.

Exercise 10

Let ! and $\oplus$ be defined as follows: a ! b = 2 and $a \oplus b$ = 3ab.

a. Write the general equation that is true if ! distributes over $\oplus$.

b. Does ! distribute over $\oplus$? _________

c. If ! distributes over $\oplus$, prove it. Otherwise, provide a counterexample to illustrate that !does not distribute over $\oplus$.

Continuation of exercise 10 where a ! b = 2 and a $\oplus$ b = 3ab.

d. Write the general equation that is true if $\oplus$ distributes over !.

e. Does $\oplus$ distribute over ! ? __________

f. If $\oplus$ distributes over ! , prove it. Otherwise, provide a counterexample to illustrate that $\oplus$ does not distribute over !.

Exercise 11

Write the general equation that is true if addition distributes over multiplication.
Does addition distribute over multiplication? _________
If addition distributes over multiplication, prove it. Otherwise, provide a counterexample to illustrate that addition does not distribute over multiplication.

Exercise 12

Write the general equation that is true if multiplication distributes over subtraction.
Does multiplication distribute over subtraction? _________
If multiplication distributes over subtraction, prove it. Otherwise, provide a counterexample to illustrate that multiplication does not distribute over subtraction.

Exercise 13

Let , and @ be defined as follows: a , $b = a^{2} + b^{2}$ and a @ b = 2b

Write the general equation that is true if , distributes over @.
Does , distribute over @? __________
If , distributes over @, prove it. Otherwise, provide a counterexample to illustrate that , does not distribute over @.
Write the general equation that is true if @ distributes over ,.
Does @ distribute over ,? __________
If @ distributes over ,, prove it. Otherwise, provide a counterexample to illustrate that @ does not distribute over ,.

Exercise 14

Make up and define two new operations.
Write a general equation that is true if one operation distributes over the other one.
Determine if the distributive property holds for your operations by proving it or providing a counterexample illustrating the equation in part b is not true.

Exercise 15

Define $\oint$ and $\boxed{\wedge}$ as follows: m $\oint$ n = 2m + 3n and m $\boxed{\wedge}$ n = mn + 2

a. State the equation that is true if $\oint$ is commutative:

Is $\oint$ commutative?