Can someone help me with this deductive reasoning and rule of inference assignment? Here are the questions, and I need help with Part b), c), and d)

Question 3. Deductive Reasoning [32 MARKS] For each of the following arguments, either: . prove the argument is valid by using Rules of Inference, OR . prove the argument is not valid by providing a world that makes the premises true and the conclusion false. Part (a) [8 MARKS] All unicorns are either white or rainbow coloured. There is a unicorn that is not white. Therefore, there is a unicorn that is rainbow coloured. Let U(x) stand for "r is a unicorn", W(x) stand for "r is white", R(x) stand for "r is rainbow coloured", universe of discourse is creatures. Part (b) [8 MARKS] All unicorns are either white or rainbow coloured. White unicorns are magical. Rainbow unicorns can fly. Only magical creatures fly. Therefore, all unicorns are magical. Let W(x) stand for "r is white", R(x) stand for "r is rainbow coloured", M(x) stands for "r is magical", F(x) stands for "r can fly", universe of discourse is unicorns. Part (c) [8 MARKS] Someone committed a crime. Every crime has a witness. A person can testify about a crime only if they witnessed it. If any person testifies about a crime, then the person who committed the crime goes to jail. Therefore, someone will go to jail. Let C(x, y) be "person a commits crime y", W(r, y) be "person r witnesses crime y", T(r, y) be "r testifies about crime y", J(x) be "r goes to jail". Part (d) [8 MARKS] Everyone who witnessed a crime must testify about it. At least one person witnessed at least one crime. If any person testifies about a crime, then the person who committed the crime is found guilty. If a crime is witnessed, it was committed by someone. Everyone who is found guilty goes to jail. Therefore, someone will go to jail. Let C(x, y) be "person a commits crime y", W(r, y) be "person r witnesses crime y", T(r, y) be "r testifies about crime y", G(x) be "r is found guilty", J(x) be "r goes to jail".Rules of Inference with Quantifiers Example: Prove the argument below is valid. Everyone likes mathematics. Everyone who likes mathematics will do well in this course. Therefore, everyone will do well in this course. Let M(x) be \"x likes mathematics\Rules of Inference Rule of Inference Tautology Name P p - (pv q) Addition p V q pq (p A q) -+ p Simplification .. P P p -+ q (PA (p -+ 9)) -+ q Modus Ponens .. q p -> q -9 ((p -+ 9) 1-9) - -p Modus Tollens -p p-+ q q -+ r ( (p -+ 9) 1 (9 7 1) ) -+ (p-+ r) Hypothetical .'. p -+ r Syllogism p V q -p ((pv q) A (-p) ) -+ q Disjunctive .'. q Syllogism P (PA q) -+ (PA q) Conjunction p q pV q -p Vr ( (pv q) A ( - pv r)) -+ (q vr) Resolution .'. q vrRules of Inference with Quantifiers Rule of Inference Name (x)d' xA Universal instantiation ... P(c) P(c) for an arbitrary c Universal generalisation (x)d'XA : 3x, P (x ) Existential instantiation .. P(c) for some element c P(c) for some element c Existential generalisation 3x, P(x) 2c must be an element in the universe of discourse