Short Answer Type
Long Answer TypeLet A = N x N and * be the binary operation on A defined by
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative. Find the identity element for *on A, if any.
Short Answer TypeConsider the binary operation A on the set {1, 2, 3, 4, 5} defined by a ∧ b = min. {a, b}. Write the operation table of the operation ∧.
Let A = N x N and let ‘*’ be a binary operation on A defined by (a, b) * (c, d) = (a c, b d). Show that
(i) (A, *) is associative (ii) (A, *) is commutative.
Long Answer TypeLet A = N x N and let ‘*’ be a binary operation on A defined by
(a, b) * (c, d) = (ad + bc, bd). Show that
(i) (A, *) is associative, (ii) (A, *) has no identity element,
(iii) Is (A, *) commutative ?
Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication 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 |
(i) Compute (2 * 3) * 4 and 2 * (3 * 4)
(ii) Is * commutative ?
(iii) Compute (2 * 3) * (4 * 5).
Let A = {1,2, 3,4, 5}
Multiplication table is given as follows :
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 |
(i) (2 *3) *4 = 1 *4 = 1
2* (3 *4) = 2* 1 = 1
(ii) Since the multiplication table is symmetrical about the diagonal starting at the upper left corner and ending at the lower right corner.
* is commutative.
(iii) 2 * 3 = 1, 4 * 5 = 1
∴ (2 * 3) * (4 * 5) = 1 * 1 = 1
Short Answer Type
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