Short Answer TypeCheck the injectivity and surjectivity of the following functions:
f : N → N given by f(x) : x2
Check the injectivity and surjectivity of the following functions.
f : Z → Z given by f(x) = x2
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 – 4x
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
Consider 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 ∧.
We construct the composition table as follows :
Table
|
* |
0 |
1 |
2 |
3 |
4 |
5 |
|
0 |
0 |
1 |
2 |
3 |
4 |
5 |
|
1 |
1 |
2 |
3 |
4 |
5 |
0 |
|
2 |
2 |
3 |
4 |
5 |
0 |
1 |
|
3 |
3 |
4 |
5 |
0 |
1 |
2 |
|
4 |
4 |
5 |
0 |
1 |
2 |
3 |
|
5 |
5 |
0 |
1 |
2 |
3 |
4 |
From this table, it is clear that
0 * 0 = 0, 1 * 0 = 0 * 1 = 1, 2 * 0 = 0 * 2 = 2, 3 * 0 = 0 * 3 = 3,
4 * 0 = 0 * 4 = 4 and 0 * 5 = 5 * 0 = 5.
0 is the identity element.
Also for each a ≠ 0 in {0, 1, 2, 3, 4, 5 },
6 – a ∈ {0, 1,2, 3, 4, 5} and a * (6 – a) = a + (6 – a) – 6 = 0.
∴ 6 – a is inverse of a for each a ≠ 0 in the set {0, 1, 2, 3, 4, 5} Also 0 * 0 = 0, i.e., 0 is inverse of itself.
Let A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b).
Then
(i) Find the identify element of (A, *)
(ii) Find the invertible elements of (A, *)
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