the operation artibitary function defined by a artibitary function b = ab divided by 7 is not a binary operation of on
(a)Q
(b)Z
(c)R
(d)C
Answers
Step-by-step explanation:
The essence to prove ∗ a binary operation is to show that ∗:S×S→S a map. In your question since ∗ is defined using multiplication and addition of R which are binary operations, we have ∗:S×S→R a map. As S=R∖{−1}, it suffices to show that the range of ∗ is S. Suppose a∗b=−1 and we see a=−1 or b=−1, a contradiction
Types of Binary Operation
Binary Addition.
Binary Subtraction.
Binary Multiplication.
Binary Division.
Definition 1. A binary operation ∗ on a set S is a function mapping S × S into S. For each (a, b) ∈ S × S, we denote ∗((a, b)) of S by a ∗ b. Example 1. Our usual addition + is a binary opera- tion on the real numbers R.
A binary operation ∗ on A is associative if ∀a, b, c ∈ A, (a ∗ b) ∗ c = a ∗ (b ∗ c). A binary operation ∗ on A is commutative if ∀a, b ∈ A, a ∗ b = b ∗ a. DEFINITION 3. If ∗ is a binary operation on A, an element e ∈ A is an identity element of A w.r.t ∗ if ∀a ∈ A, a ∗ e = e ∗ a = a.
Answer:
∗b=
b+1
a
b∗a=
a+1
b
b+1
a
=
a+1
b
⇒a∗b
=b∗a
∴∗ is not commutative
a∗(b∗c)=a∗[
c+1
b
]=
c+1
b
+1
a
=
b+c+1
a(c+1)
(a∗b)∗c=(
b+1
a
)∗c=
c+1
b+1
a
a=
(b+1)(c+1)
a
a∗(b∗c)
=(a∗b)∗c
∴∗ is neither associative nor commutativea