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.
6 answers
Please work out the answer for me.
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
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
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Mathematics
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)
0 answers