UNIT 3 : Binary operations
Key unit competence
Use mathematical logic to understand and perform operations using the properties of algebraic structures.
Learning objectives
3.1 Introduction
Activity 3.1
Work in groups.
1. Discuss what you understand by the term binary operations.
2. Carry out research to find the meaning as used in mathematics.
In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set. Typical examples of binary operations are the addition (+) and multiplication (×) of numbers and matrices as well as composition of functions on a single set. For example:
• On the set of real numbers , f(a, b) = a × b is a binary operation since the sum of two real numbers is a real number.
• On the set of natural numbers , f(a, b) = a + b is a binary operation since the multiplication of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different.
3.2 Groups and rings
Activity 3.2
Work in groups.
1. Discuss what you understand by the terms binary, group, ring, integral domain and field.
2. Carry out research to find their meanings as used in mathematics.
Group
A group (G,*) is a non-empty set (G) on which a given binary operation (*) is defined such that the following properties are satisfied:
(a) Closure property: ∀ a, b ∈ G, (a * b) ∈ G.
(b) Associative property: ∀ a, b, c ∈ G, a * (b * c) = (a* b) *c
(c) Identity property: ∀ a ∈ G, ∃ e ∈ G, a * e = e * a = a
(d) Inverse element property : ∀ a ∈ G, ∃ a–1 ∈ G, a * a–1 = a–1 * a = e, where e is an identity element.
A group is commutative (Abelian) if in addition to the previous axioms, it satisfies commutativity: ∀ a, b ∈ G, a * b = b * a
Subgroups
A non-empty subset H of a group G is a subgroup if the elements of H form a group under the operation from G restricted to H. The entire group is a subgroup of itself and is called the improper subgroup.
Every group has a subgroup consisting of an identity element alone and is called the trivial subgroup. The identity element is an element of every subgroup of a group.
If H ≠ G, we call it a subgroup H of G; proper, and we write H < G.
If H ≠ {e}, we call it a subgroup H of G; nontrivial, and we write H ≤ G.
(H, ) is a subgroup of (G, ) if it verifies the following conditions:
1) Closure: ∀ x, y, ∈ H, (x y) ∈ H
2) e ∈ H
3) ∀ x, ∈ H, x–1 ∈ H
where e is the identity element and x–1 is the inverse of x.
Rings
3.3 Fields and integral domains
Fields
A field F is a non-empty set defined by the following properties. For all a, b, c ∈ F:
Integral domain
An integral domain is a commutative ring with an identity (1≠ 0) with no zerodivisors. That is ab = 0 ⇒ a = 0 or b = 0.
3.4 Cayley tables
A Cayley table is a useful device for studying binary operations on finite sets. It can be used to help determine if the set under a given operation is a group or not. Given a binary operation (S, *), with S a finite set, its Cayley table has the elements of S listed in the top row and left hand column of the table, and inside the table we write the outcome a * b of the operation in the row labelled with a and the column labelled with b. The elements should be placed along the top row and the left side column in the same order. The following illustrates the Cayley table of the set S = {a, b} under the operation *:
Below we use Cayley tables to illustrate the groups of order 2, 3 and 4.
We remark that a * a = a, a * b = b, b * a = b, and b * b = a
We can use Cayley tables to determine if (S, *) is closed, commutative, admits identity element or inverse element.
Closure: If all elements inside the table are in the original set S, then (S, *) is closure.
Commutative: The Cayley table is symmetric about the diagonal. This will only happen if every corresponding row and column are identical.
Identity: In a Cayley table, the identity element is the one that leaves all elements of the set S unchanged.
Inverse: It is found by asking: ‘’What other element can I combine with this one to get the identity?’’
Associative: It is checked by simply respecting the rules of parentheses in comparing if the two sides are equal.
Modular arithmetic
