Test Your Understanding about Binary Operations PART 1

Algebra Level 2

In the algebra of numbers, we are primarily concerned with four binary operations: Addition, Multiplication, Subtraction, and Division. And a binary operation is a rule that associates with an ordered pair of elements a third unique element. A binary operation is defined on some set $S;$ that is: The ordered pair of elements belong to the same set $S.$

Consider the binary operation of addition defined on the set $W$ of whole numbers.
The operation associates with every ordered pair of elements, a unique third number denoted as $a + b.$

Now Test Your Understanding about BINARY OPERATIONS PART 1

Let $D$ be the set of odd whole numbers.

$D = \{ 1,3 ,5,7, \ldots \}$

Is addition a binary operation on the set $D?$

It isn't! It is!

2 solutions

Tom Engelsman
Apr 28, 2021

If $D = [x = 2n+1 : n \in \mathbb{N_{0}}]$ , then addition of any two distinct elements in $D$ yields $(2p+1) + (2q+1) = 2(p+q) + 2 = 2(p+q+1)$ . The result is an even number which can never belong to $D \Rightarrow$ addition is NOT a binary operation in this set.

Mr Mokelu
Apr 27, 2021

By definition, a binary operation must associate with every ordered pair of numbers, a unique third number a + b.

But 1 + 3 = 4, therefore addition is not a binary operation on the set D.

Test Your Understanding about Binary Operations PART 1

2 solutions

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