Advertisements
Advertisements
प्रश्न
Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the closure, commutative and associate properties satisfied by * on Q.
उत्तर
Given a * b = `(("a" + "b")/2)`; a, b ∈ Q
a ∈ Q and b ∈ Q
⇒ a * b = `("a" + "b")/2` ∈ Q
Hence * is a binary operation on Q
a * b = `("a" + "b")/2`
b * a = `("b" + "a")/2`
= `("a" + "b")/2` .......[∵ a + b = b + a]
∴ Binary operation * is commutative
a * (b * c) = a * `(("b" + "c")/2)`
= `("a" + ("b" + "c")/2)/2`
= `(2"a" + "b" + "c")/2`
(a * b) * c = `(("a" + "b")/2)` * c
= `("a" + "b" + 2"c")/4`
So, a * (b * c) ≠ (a * b) * c
Hence, the binary operation * is not associative.
APPEARS IN
संबंधित प्रश्न
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = ab
Let A = Q x Q and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ∈ A. Determine, whether * is commutative and associative. Then, with respect to * on A
1) Find the identity element in A
2) Find the invertible elements of A.
Check the commutativity and associativity of the following binary operation '*'. on Z defined by a * b = a + b + ab for all a, b ∈ Z ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = a + ab for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on R defined by a * b = a + b − 7 for all a, b ∈ R ?
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].
Show that 'o' is both commutative and associate ?
Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
Find the invertible element in A ?
Let A \[=\] R \[\times\] R and \[*\] be a binary operation on A defined by \[(a, b) * (c, d) = (a + c, b + d) .\] . Show that \[*\] is commutative and associative. Find the binary element for \[*\] on A, if any.
Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.
Write the multiplication table for the set of integers modulo 5.
Define a commutative binary operation on a set.
Define identity element for a binary operation defined on a set.
Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.
Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?
Let N be the set of natural numbers. Then, the binary operation * in N defined as a * b = a + b, ∀ a, b ∈ N has identity element.
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = (a – b)2 ∀ a, b ∈ Q
If the binary operation * is defined on the set Q + of all positive rational numbers by a * b = `" ab"/4. "Then" 3 "*" (1/5 "*" 1/2)` is equal to ____________.
The identity element for the binary operation * defined on Q – {0} as a * b = `"ab"/2 AA "a, b" in "Q" - {0}` is ____________.
Let * be a binary operation on the set of integers I, defined by a * b = a + b – 3, then find the value of 3 * 4.
