Question

If \( A = \{1, 2, 3\} \), \( B = \{4, 5, 6, 7\} \), and \( f = \{(1, 4), (2, 5), (3, 6)\} \) is a function from A to B, then show that \( f \) is one-one.

Hint:

For a function to be one-one, each element of the domain must map to a unique element of the codomain.

Answer 1

Step 1: Understanding one-one function.
A function \( f: A \to B \) is one-one (or injective) if distinct elements in the domain \( A \) map to distinct elements in the codomain \( B \). That is, if \( f(a_1) = f(a_2) \), then \( a_1 = a_2 \).

Step 2: Check the given function.
The function \( f \) is given by: \[ f = \{(1, 4), (2, 5), (3, 6)\} \] We observe that: - \( f(1) = 4 \) - \( f(2) = 5 \) - \( f(3) = 6 \) There is no repetition of values in the range (i.e., 4, 5, and 6 are distinct). Therefore, \( f \) is one-one.

Step 3: Conclusion.
Since all elements of \( A \) map to distinct elements in \( B \), the function \( f \) is one-one.