DSA: If signature is accepted then signature is correct
This isn't completely standard terminology, so you should check the precise definitions in your lecture notes. But I can't think of anything else that the exercise should be.

You have a definition of the DSA signature process: given some parameters $(p,q,g)$, a private key $x$ and a message $m$, generate a nonce $k$ and calculate $(r,s)$ given by a certain formula. You also have a definition of the DSA verification process: given some parameters $(p,q,g)$, a public key $y$, a message $m$ and a candidate signature $(r,s)$, make a certain calculation and output “accepted” or “rejected”. This is a signature scheme if every output of the signature process is accepted by the verification process, that is, if you take the $(r,s)$ given by signing and perform the verification process on it, the output is “accepted”.

This exercise asks you to prove a dual property which is a practical necessity, while not sufficient for security: given parameters $(p,q,g)$, a key pair $(x,y)$ and a message $m$, if the verification process for $y$, $m$ and the signature candidate $(r,s)$ outputs “accepted”, then there exists a nonce $k$ such that the signature process for $x$ and $m$ yields the output $(r,s)$.


   The calculation itself is easy: take $k = s^{-1} (m + x\,r)$, inverting the formula used to compute $s$ from $k$ and $r$ during the signature process. The verification process basically checks that $g^k = r$, so if the signature is accepted, that means that it's the output of the signature process for this $k$.


This property is practically necessary for security because if the adversary can efficiently find $(r,s)$ which is not the output of the signing process, that means that they can craft an invalid signature. I'm not sure if the mere existence of a valid signature that no one can find efficiently would disqualify the signature scheme.