Question

If $ A=\left\{ 1,2,3 \right\} $ , show that a one-one function $ f:A\to A $ must be onto.

Answer 1

Hint: We first try to explain the formulas for one-one and onto. We then explain the image and preimage concept and try to find the mappings for the set A. We prove that the one-one function $ f:A\to A $ must be onto.

Complete step by step solution:
It’s given $ A=\left\{ 1,2,3 \right\} $ . We are taking the one-one function $ f:A\to A $ .
The function is mapping the points to itself.
For every point in A there will be only one image in A.
No two points in A will map to the same point in A as then the function will not be one-one.
Therefore, when we take a point in the image form of A, we will get only one preimage in A.
This proves that we have the function $ f:A\to A $ to be onto.
For example; if we take $ f\left( 1 \right)=2,f\left( 3 \right)=1,f\left( 2 \right)=3 $ in $ f:A\to A $ , then we get that the function is also onto as for every image in A, we have only one preimage in A.
Thus, proved a one-one function $ f:A\to A $ must be onto.

Note: We prove that the mathematical expression for one-one function is that if \[f\left( a \right)=f\left( b \right)\] then we get $ a=b $ . We can also use this to prove that the one-one function $ f:A\to A $ must be onto.