Lecture 06 Part 4

Binary Operations

Operations like addition, subtraction, multiplication and division are some of the very first lessons a child learns in mathematics -- next, perhaps, to counting. These operations are technically called binary operations because they "operate two numbers".

Algebraic Definition of Binary Operations

Binary operations can be thought like functions mapping a pair of numbers into another number.

For example, addition is a function that maps from ℝ² → ℝ which is expressed as +(a,b) (a function that has an ordered pair as its domain). However, instead of the functional notation above, the more popular notation is of course, a + b. For instance,

+(4,2) = 4 + 2 = 6

So formally, a binary operation on a set S is a mapping

S² → T.

In higher mathematics, there are operations that operate more than two numbers. However, in this course, we will only cover binary ones. Hence, we will just say operation to mean binary operation.

There are four basic operations in mathematics: addition, subtraction, multiplication, and division. There is a basic rule in evaluating these operations. This rule is popularly called "PEMDAS".

Now let's talk about the viral meme:

Again, mathematical language should be precise!

Regardless of what the answer to the problem is, we should avoid writing expressions in this way. As emphasized in Lecture 04 at the beginning of this chapter, mathematical language should be precise. It is supposedly clear and not vague. If a mathematical expression causes confusion, then it better be rewritten in a way that is better understood.

Properties of Binary Operations

Some operations exhibit interesting properties. These properties such as closure, commutativity, associativity, identity, inverse, and distributivity are sometimes helpful in evaluating operations and in proving.

We will use the symbol * to represent any operation.

Closure Property

An operation * on a set S is well-defined if and only if for any a, b ∈ S, then

a * b ∈ S.

In turn, the set S is said to be closed under the operation *.

For example, addition, subtraction, and multiplication are all well-defined in the set of real numbers because when we add, subtract or multiply any two real numbers, the answer is still a real number. However, division is not well-defined because when we divide a number by zero, the quotient is either indeterminate (0 ÷ 0) or undefined (a ÷ 0 for a ≠ 0). So we can also say that ℝ is closed under addition, subtraction, and multiplication.

Note that being well-defined depends on the set where the operation takes place. So it is possible that an operation may be well-defined over a given set but not in another. For example, multiplication is well-defined over the set of real numbers, but not over the set of negative real numbers because multiplying two negative numbers is positive. So the set of negative real numbers is not closed under multiplication.

Exercise. Is division well-defined over the following sets?

real numbers
positive real numbers
negative real numbers
positive integers
positive rational numbers

Commutative Property

An operation * is commutative if and only if

a * b = b * a.

In other words, an operation is commutative if and only if the order of which the numbers are written does not matter. Obviously, addition and multiplication are commutative. However, subtraction and division are not.

Exercise.

Define the operation ⊕ on ℝ by a ⊕ b = a² + b². Is ⊕ commutative? Illustrate your answer by showing two examples.
Define ⊖ by a ⊖ b = b² - a². Is ⊖ commutative? Illustrate your answer by showing two examples.

Associative Property

An operation * is said to be associative if and only if

(a * b) * c = a * (b * c).

In other words, an associative operation is a binary operation in which the grouping does not matter. Addition and multiplication are associative. In a series of additions or a series of multiplications, it does not matter which pair you treat first. The answer will be the same.

Exercise.

Is subtraction associative? How about division?
Define ⊖ and ⊕ as above. Which one is associative?

Identity Property

Let * be an operation on a set S. An identity element in S under * is an element e such that

a * e = a
e * a = a

Some operations have an identity element while others do not. Those with an identity element are said to satisfy the identity property. Examples of this are addition and multiplication. The identity element for addition (may also be called the additive identity) is zero (0):

a + 0 = a
0 + a = a

The identity element for multiplication (may also be called the multiplicative identity) is one (1):

a × 1 = a
1 × a = a

There are also some operations, especially those which are not commutative, in which the identity element works only in one place. For example, zero is an identity element for subtraction but only when it is located right of the minus sign (i.e., when zero is the subtrahend):

a - 0 = a
0 - a = -a

This is an example of a right identity element. A right identity element is an identity element but works only when it is located at the right of the binary operation. A left identity element is defined similarly as an identity element that works when it is at the left side of the operation. Take note that an element can be called an identity ONLY WHEN it is both a right and a left identity element. Otherwise, we should specify.

Exercise.

Does division have an identity element?
Define ⊖ and ⊕ as above. Do they have identity elements? If yes, what are these?

Inverse Property

Suppose * is an operation on a set S with identity element e. Also, let a be an element of S. We say that an element b is an inverse of a if and only if

a * b = e
b * a = e

One important thing to note first is that an inverse element depends on an identity element. In other words, there is no inverse element if there is no identity element for the operation. However, when a a set has an identity for the operation, it does not necessarily imply that there are inverse elements in it. An inverse element is an element such that when they are both operated, results to the identity.

For example, the inverse element of a under addition is -a:

a + (-a) = 0
-a + a = 0

Thus, the negative of a number may also be called its additive inverse.

For multiplication, the inverse of a is its reciprocal:

While it is true that reciprocals, also called multiplicative inverses, are the inverse elements under multiplication, not all numbers have an inverse. Specifically, the number zero does not have a multiplicative inverse (because 1/0 is undefined). The lesson here is that even if there is an identity for an operation, and there are inverses for some elements, it does not mean that all elements have an inverse.

Exercise. Are there inverses for real numbers under ⊕ above?

Distributive Property

When there are two elements defined in one set, some exhibit the distributive property. For example, we have the distributive property of multiplication over addition:

a (b + c) = ab + ac

Exercise. Define ⊞ and ⊙ on the set of real numbers as follows:

Evaluate

6 ⊙ 8
6 ⊙ 4
6 ⊞ 6

Is ⊙ distributive over ⊞? Show an example.

References

[1] Fraleigh, J. B. (2003). A first course in abstract algebra. Pearson Education India.
[2] Dummit, D. S., & Foote, R. M. (2004). Abstract algebra (Vol. 3). Hoboken: Wiley.
[3] Nicholson, W. K. (2012). Introduction to abstract algebra. John Wiley & Sons.

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