PLEASE NEED HELP WITH THESE

4) [20 marks total] Assume that R denotes "Roses are red", V denotes "Violets are blue", and S denotes "Sugar is sweet". Translate the following statements into symbolic notation using compound propositions. a.[5 marks] Roses are red, and violets are blue. b[5 marks] Roses are red, and either violets are blue, or sugar is sweet. c.[5 marks] Roses are red only if violets aren't blue or sugar is sour. d.[5 marks] Roses are red and, if sugar is sour, then either violets aren't blue, or sugar is sweet. 5) [20 marks total] Consider the statement: "My brother is left-handed, but if the key is not missing, then my brother is not left-handed; therefore, the key is missing." a. [10 marks] Assign letters to the basic propositions and write each hypothesis and the conclusion as an implication. b. [10 marks] Prove the correctness of your answer to item a. by using the logical equivalencies and/or rules of inferences tables from your text which are also available below. 6) [20 marks total] Consider the domain of people and the following argument: Hypothesis: - If a person is a mathematician, then he/she is a programmer. - If a person is a programmer, then he/she is prone to headaches. Conclusion: - Therefore, if a person is a mathematician, then he/she is prone to headaches. 1. [10 marks] Assign letters to the basic propositions and write each hypothesis and the conclusion as an implication. Hint: Use the syntax P(X)"X>0 " or P(X) : "subject and predicate" to represent your statements. 2. [10 marks] Prove the correctness of you answer to item a. by using the logical equivalencies and/or rules of inferences tables from your text which are also available below