1
Question
Consider
a binary operation * on
the set {1, 2, 3, 4, 5} given by the following multiplication table.
(i) Compute
(2 *
3) * 4
and 2 * (3
* 4)
(ii) Is
* commutative?
(iii) Compute
(2 * 3)
* (4
* 5).
(Hint:
use the following table)
*
1
2
3
4
5
1
1
1
1
1
1
2
1
2
1
2
1
3
1
1
3
1
1
4
1
2
1
4
1
5
1
1
1
1
5
Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.
(i) Compute (2 * 3) * 4 and 2 * (3 * 4)
(ii) Is * commutative?
(iii) Compute (2 * 3) * (4 * 5).
(Hint: use the following table)
|
* |
1 |
2 |
3 |
4 |
5 |
|
1 |
1 |
1 |
1 |
1 |
1 |
|
2 |
1 |
2 |
1 |
2 |
1 |
|
3 |
1 |
1 |
3 |
1 |
1 |
|
4 |
1 |
2 |
1 |
4 |
1 |
|
5 |
1 |
1 |
1 |
1 |
5 |
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Solution
(i) (2 *
3) *
4 = 1 *
4 = 1
2 *
(3 *
4) = 2 *
1 = 1
(ii) For every a, b ∈{1,
2, 3, 4, 5}, we have a *
b = b *
a. Therefore, the operation *
is commutative.
(iii) (2 *
3) = 1 and (4 *
5) = 1
∴(2 *
3) *
(4 *
5) = 1 *
1 = 1
(i) (2 * 3) * 4 = 1 * 4 = 1
2 * (3 * 4) = 2 * 1 = 1
(ii) For every a, b ∈{1, 2, 3, 4, 5}, we have a * b = b * a. Therefore, the operation * is commutative.
(iii) (2 * 3) = 1 and (4 * 5) = 1
∴(2 * 3) * (4 * 5) = 1 * 1 = 1
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