Answer to Question #165322 in Discrete Mathematics for Faheem

Question

 A f:X→Y where X,Y ∈ Z⁺ , and  0≤ X ≤23 and, 0≤Y≤59, X is hour of clock and Y is minute of digital clock, Proof the following

                                                       a.     Is f:X→Y is well defined

b.     Is f:X→Y a one-to-one correspondence 

Answer 1

Let $f:X\to Y$ where $x, y\in \mathbb Z^+$, and $0\le x\le 23, 0\le y\le 59$, $x$ is hour of clock and $y$ is minute of digital clock.

a.     Let us show that $f:X→Y$ is not well defined. Indead, if we consider $x=1$ and $y_1=10,\ y_2=20$, that is the hours $1:10,\ 1:20$, we have that $x=1$ maps to at least two values $y=10$ and $y=20.$ Therefore, $f:X→Y$ is not well defined.

b.   Let us show that $f:X→Y$ is not a one-to-one correspondence. Indeed, if we consider $x_1=1$ and $x_2=2,\ y_1=y_2=20$, that is the hours $1:20,\ 2:20$, we have that $x_1=1$ and $x_2=2$ map to the same value $y=20.$ Therefore, $f:X→Y$ is not a one-to-one correspondence.