1
Question
Is
* defined
on the set {1, 2, 3, 4, 5} by a
* b
= L.C.M.
of a and
b a
binary operation? Justify your answer.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. of a and b a binary operation? Justify your answer.
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Solution
The
operation *
on the set A = {1, 2, 3, 4, 5} is defined as
a *
b = L.C.M. of a and b.
Then, the
operation table for the given operation *
can be given as:
*
1
2
3
4
5
1
1
2
3
4
5
2
2
2
6
4
10
3
3
6
3
12
15
4
4
4
12
4
20
5
5
10
15
20
5
It can be
observed from the obtained table that:
3 *
2 = 2 *
3 = 6 ∉ A, 5 *
2 = 2 *
5 = 10 ∉ A, 3 *
4 = 4 *
3 = 12 ∉ A
3 *
5 = 5 *
3 = 15 ∉ A, 4 *
5 = 5 *
4 = 20 ∉ A
Hence, the
given operation *
is not a binary operation.
The operation * on the set A = {1, 2, 3, 4, 5} is defined as
a * b = L.C.M. of a and b.
Then, the operation table for the given operation * can be given as:
|
* |
1 |
2 |
3 |
4 |
5 |
|
1 |
1 |
2 |
3 |
4 |
5 |
|
2 |
2 |
2 |
6 |
4 |
10 |
|
3 |
3 |
6 |
3 |
12 |
15 |
|
4 |
4 |
4 |
12 |
4 |
20 |
|
5 |
5 |
10 |
15 |
20 |
5 |
It can be observed from the obtained table that:
3 * 2 = 2 * 3 = 6 ∉ A, 5 * 2 = 2 * 5 = 10 ∉ A, 3 * 4 = 4 * 3 = 12 ∉ A
3 * 5 = 5 * 3 = 15 ∉ A, 4 * 5 = 5 * 4 = 20 ∉ A
Hence, the given operation * is not a binary operation.
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