A binary operation * is defined on the set R of real numbers by: a+b+ab where a, b€R .Calculate 5*(-2)*5.Find the identity element of R under the operation*. Determine the inverse under * of a general element a €R
This is another way to solve it:
2.b = a + b + ab where a, b ER
5.(-2).5
solution
5+(-2)=5+(-2)+5X(-2)
= 5-2-10
= 3-10 =-7
(-7).5=- 7+5+(-7) 5
= 7+5 35
=-2-35
=-37
(ii) a. era
ate+aera
etae = a-a
e(1+a)=0
e = 0
Let a' be the inverse of a, then
ata' + aa' = 0
a' taa' =-a
a'(1+a)=-a
I shall assume that * associates left to right, like addition. If so, then
5*(-2)*5 = (5*(-2))*5 = (5 + -2 + 5(-2))*5 = (-13)*5
= -13 + 5 + (-13)(5) = -73
What is the identity element under *? It must be zero, since a*0 = a
If b = a-1 under * then a*b = b*a = 0
So, what do you think a-1 is?
Respond
first 5×-2+(5+-2) then...
A binary operation * is defined on the set R of real number by a * b = a + b + ab ( where a, b belong to R ). Calculate 5 * ( - 2 ) *, and find the identity element e of R under the operation. Then determine the inverse under * of a general element a belong to R and state which element has no inverse.
Answer to the question
oops. 5*-2 = -7
fix the 2nd step.
Akingbade is Correct
but is the remaining part
a^-1(1+a)=~a
since we are looking for a^-1 then we make a^-1 the subject of Formula
a^-1=-a/(1+a)
Ogunremi
To calculate 5*(-2)*5 using the binary operation *, we substitute a = 5 and b = -2 into the given expression: 5 + (-2) + 5*(-2).
Step 1: Multiply 5 by -2: 5*(-2) = -10.
Step 2: Substitute the value of -10 into the expression: 5 + (-2) + (-10).
Step 3: Add the numbers: 5 + (-2) + (-10) = -7.
Therefore, 5*(-2)*5 = -7.
To find the identity element of the set R under the operation *, we need to find an element e ∈ R such that for any a ∈ R, a*e = a and e*a = a.
Let e be the identity element. Substituting these values into the given expression, we have: a + e + ae = a, and e + a + ae = a.
Using a = 0, we get: e = 0 + e + 0*e, and e + 0 + 0*e = 0.
Simplifying these equations, we have: e = e + 0, and e = 0 + e.
This implies that the identity element e is 0.
To find the inverse of a general element a ∈ R under the operation *, we need to find an element b ∈ R such that a * b = e and b * a = e, where e is the identity element (which we already determined to be 0).
Using the expression for the operation *, a + b + ab, we substitute e = 0 and simplify:
a + b + ab = 0
Rearranging the equation, we get:
a + ab = -b
Factoring out b, we have:
b(a + 1) = -a
Dividing both sides by (a + 1), we obtain:
b = -a / (a + 1)
Thus, the inverse of a general element a ∈ R under the operation * is -a / (a + 1).