Here is how you can show closure under addition and multiplication for the first problem:RingClosure
Some courses seem to require that closure under those operations be demonstrated, while others don't.
Regarding the other task, I don't understand how you are supposed to "prove the definition of injective and surjective". For that matter, I don't understand how one is supposed to "prove" the definition of anything. A definition is simply an agreed upon explanation of an idea or concept. The definition itself isn't something that can be proved. Perhaps there is some confusion over terminology here.
I don't know what I can add to that part. The definition of an injective function is that each value in the range of the function corresponds to exactly one element of the domain. This means that there cannot be two or more values in the domain that map to the same value in the range. In other words, each (a, f(a)) pair is a unique combination of a value from the domain and a value from the range. With only five values in Z5 and G in the proof, this can be seen by inspection, since every value from G appears once and only once on the right side of the equations that are listed for phi(n) on page 4 of the previous document. As noted in the earlier solution, each element of Z5 maps to one and only one value in G, and it maps to a unique value in G (meaning that no two values from Z5 map to the same value in G).
The definition of a surjective function is that each value in the range is mapped to by some value in the domain. In other words, there are no values in the range which DO NOT have a corresponding value in the domain. Another way of thinking about this is that the range is "covered"
by the domain. Again, with Z5 and G, this can easily be seen since each value in G appears on the right side of an equation on page 4.