Question
Download Solution PDFIf * is a binary operation on Z such that a * b = a + b + 1 ∀ a, b ∈ Z then find the identity element of Z with respect to * ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Let * be a binary operation on a non-empty set S. If there exists an element e in S such that a * e = e * a = a ∀ a ∈ S. Then the element e is said to be an identity element of S with respect to *.
Calculation:
Given: * is a binary operation on Z such that a * b = a + b + 1 ∀ a, b ∈ Z.
Let e be the identity element of Z with respect to *.
As we know that if e is an identity element of a non-empty set S with respect to a binary operation * then a * e = e * a = a ∀ a ∈ S.
Let a ∈ Z and because e is the identity element of Z with respect to given operation *.
i.e a * e = a = e * a ∀ a ∈ Z.
According to the definition of *, we have
⇒ a * e = a + e + 1 = a ∀ a ∈ Z
⇒ e = - 1
Hence, - 1 is the identity element of Z with respect to given operation *.
Last updated on May 26, 2025
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