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. Determine the inverse under * of a general element a belong to R and state which element has no inverse.
a * b = a + b + ab
5 * ( - 2 ) = 5 + (-2) + (5)(-2) = -7
If e is the identity element the e*b = b
but e*b = e + b + eb
It follows that e + b + eb = b
e + eb = 0
e(1 + b) = 0
e = 0 or b = -1
check:
0*15 = 0 + 15 + 0(15) = 15
7*-1 = 7 - 1 -7 = -1
Since clearly a*b = b*a
the identity element is either 0 or -1
Please work out the answer for me.
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Determine whether each of the following operations defines a binary operation or not on the given set.( ⊕ on ℕ defined by 𝑎 ⊕ 𝑏 = 𝑎 𝑏 + 𝑏 𝑎 for all 𝑎, 𝑏 ∈ ℕ)
Determine whether each of the following operations defines a binary operation or not on the given set.( ∗ on ℕ defined by 𝑥 ∗ 𝑦 = 𝑥 + 𝑦 − 2 for all 𝑥, 𝑦 ∈ ℕ)
To calculate 5 * (-2), we substitute a = 5 and b = -2 into the expression for the binary operation *:
5 * (-2) = 5 + (-2) + 5*(-2)
= 5 - 2 - 10
= -7
So, 5 * (-2) = -7.
Now let's find the identity element e of R under the operation *. The identity element e is an element of R that, when combined with any other element a of R, gives back a. Mathematically, for any a in R:
a * e = a
Substituting a = 5 into the expression, we have:
5 * e = 5 + e + 5e
To find the identity element, we need to solve this equation. By comparing the coefficients of e, we get:
1 + 5e = 0
Solving for e, we have:
5e = -1
e = -1/5
So, the identity element e of R under the operation * is -1/5.
Next, let's determine the inverse under * of a general element a belonging to R. The inverse of a, denoted by a', is the element which, when combined with a, gives back the identity element e. Mathematically:
a * a' = e
Substituting a into the expression, we have:
a * a' = a + a' + aa'
To find the inverse element a', we need to solve this equation. By comparing the coefficients of a' and the constant term, we get:
1 + a' = 0
aa' = 0
From the first equation, we find:
a' = -1
From the second equation, we find that either a = 0 or a' = 0. If a = 0, then a' = -1/5, which is not the identity element. So, a' cannot equal 0. Therefore, the only element in R that has no inverse under the operation * is 0.
To summarize:
- 5 * (-2) = -7
- The identity element e of R under the operation * is -1/5.
- The inverse under * of a general element a belonging to R is a' = -1.
- The element 0 has no inverse under the operation * in R.