Name: Properties of Binary Operations
Uploaded: 2019-07-19
Duration: 12 min 41 s
Description: In this video, we will learn about binary operations. We will also understand properties followed by binary operations with the help of suitable examples.

Question

Let $S$ be the set of all real numbers except

- 1

and let

' *'

be an operation defined by

a * b = a + b + a b

for all

a, b \in S

. Determine whether

' *'

is a binary operation on $S$ . If yes, check its commutativity and associativity. Also, solve the equation

(2 * x) * 3 = 7

.

Accepted Answer

Given $a * b = a + b + a b$ for all $a, b \in S$ .

$S$ be the set of all real numbers except $- 1$ .

Checking for binary operation:

Let $a, b \in S$ . Then

$(2 * x) * 3 = 7$

$a * b = a + b + a b$

We need to prove that $a + b + a b \in S$ .

For this we have to prove that $a + b + a b \in R$ and $a + b + a b \neq - 1$

Since $a, b \in R$ , $a + b + a b \in R$ , let us assume that $' *'$

$a + b + a b + 1 = 0$

$' *'$

$\Rightarrow a (1 + b) + 1 (1 + b) = 0$

$\Rightarrow (a + 1) (b + 1) = 0$

$a = - 1, b = - 1$ , which is not possible.

$a + b + a b \neq - 1$

Therefore,

$a + b + a b \in S$

Thus, $*$ is binary operation on $S$ .

$*$ is commutative if $a * b = b * a$

Commutativity:

Let $a, b \in S$ . Then,

$a * b = a + b + a b$

$= b + a + b a$

$= b * a$

Therefore,

$a * b = b * a, \forall a, b \in S$

Thus, $*$ is commutative on $S$ .

$*$ is associative if $(a * b) * c = a * (b * c)$

Associativity:

Let $a, b, c \in S$

$a * (b * c) = a * (b + c + b c)$

$= a + b + c + b c + a (b + c + b c)$

$= a + b + c + b c + a b + a c + a b c$

$(a * b) * c = (a + b + a b) * c$

$= a + b + a b + c + (a + b + a b) c$

$= a + b + a b + c + a c + b c + a b c$

Therefore,

$a * (b * c) = (a * b) * c, \forall a, b, c \in S$

Thus, $*$ is associative on $S$ .

Now,

Given: $(2 * x) * 3 = 7$

$\Rightarrow (2 + x + 2 x) * 3 = 7$

$\Rightarrow (2 + 3 x) * 3 = 7$

$\Rightarrow 2 + 3 x + 3 + (2 + 3 x) 3 = 7$

$\Rightarrow 5 + 3 x + 6 + 9 x = 7$

$\Rightarrow 12 x + 11 = 7$

$\Rightarrow 12 x = - 4$

$\Rightarrow x = \frac{- 4}{12}$

$\Rightarrow x = \frac{- 1}{3}$

Let $S$ be the set of all real numbers except $- 1$ and let $' '$ be an operation defined by $a b = a + b + a b$ for all $a, b \in S$ . Determine whether $' '$ is a binary operation on $S$* . If yes, check its commutativity and associativity. Also, solve the equation $(2 * x) * 3 = 7$ .

Important Questions on Binary Operations

Learn and Prepare for any exam you want !

Let S be the set of all real numbers except −1 and let '*' be an operation defined by a*b=a+b+ab for all a,b∈S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 *x) * 3=7.

Important Questions on Binary Operations

Learn and Prepare for any exam you want !

Let $S$ be the set of all real numbers except $- 1$ and let $' '$ be an operation defined by $a b = a + b + a b$ for all $a, b \in S$ . Determine whether $' '$ is a binary operation on $S$* . If yes, check its commutativity and associativity. Also, solve the equation $(2 * x) * 3 = 7$ .