GATE CS 2015 Set 2 Q54 — Sets, Relations and Functions

Let the size of the sets be $|X| = 2$ and $|Y| = 20$.

Step 1: Total number of functions ($|F|$)
Any function $f: X \to Y$ maps each of the $2$ elements of $X$ to any of the $20$ elements of $Y$.
$$\text{Total functions} = |Y|^{|X|} = 20^2 = 400$$

Step 2: Number of one-to-one (injective) functions
For a function to be one-to-one, distinct elements in $X$ must map to distinct elements in $Y$.
- The first element of $X$ can be mapped to any of the $20$ elements in $Y$.
- The second element of $X$ can then be mapped to any of the remaining $19$ elements in $Y$.
$$\text{Number of one-to-one functions} = 20 \times 19 = 380$$

Step 3: Probability calculation
Since $f$ is chosen uniformly at random from $F$, the probability that $f$ is one-to-one is:
$$P = \frac{\text{Number of one-to-one functions}}{\text{Total number of functions}} = \frac{380}{400} = \frac{19}{20} = 0.95$$