A binary operation * is defined on the set R of real numbers by:a*b=a+b+ab where a, b € R.Calculate 5*(-2)*5. Find the identity element if R under the operation *. Determine the inverse under * of a general element a € R.
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To calculate 5*(-2)*5 using the binary operation *, substitute the values of a and b into the definition.
a * b = a + b + ab
For this expression, let's set a = 5 and b = -2.
5 * (-2) = 5 + (-2) + 5*(-2)
Simplifying the expression:
5 * (-2) = 5 - 2 - 10
5 * (-2) = -7
Therefore, 5 * (-2) * 5 = -7 * 5 = -35.
Now let's move on to finding the identity element if R under the operation *.
The identity element, denoted by e, is the element such that a * e = a = e * a for all a in R.
Let's substitute a = 5 into the expression and solve for e:
5 * e = 5 + e + 5e
Combining like terms:
6e + 5 = 5 + e
Next, subtract e from both sides:
6e = 5
Finally, divide both sides by 6:
e = 5/6
Therefore, the identity element under the operation * in R is 5/6.
Lastly, let's find the inverse under * of a general element a in R.
The inverse of an element a, denoted by a^(-1), is the element such that a * a^(-1) = e = a^(-1) * a.
We'll substitute a for an arbitrary value and solve for a^(-1):
a * a^(-1) = a + a^(-1) + aa^(-1)
Combining like terms:
a^2 + a^(-1) = a + a^(-1) + aa^(-1)
Next, we subtract aa^(-1) and a^(-1) from both sides:
a^2 = a
Now, divide both sides by a:
a = 1
Therefore, the inverse of any element a in R under the operation * is 1.