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Question \# 1: (a) Let a,b,c and d be the propositions a: The election is decided. b: The votes have been counted._ c: Democrats candidate have lost. Write these propositions using a,b, and c and logical connectives (including negations): (i) Democratic candidate have not lost if the votes have not been counted. (ii) The election is not decided only if the votes are not counted. (iii) The votes have been counted but the Democratic candidate has lost. (b) Using the premises (statements) from part (a), apply rules of inference to obtain conclusion(s) from these premises. Also, name the rules of inference used to obtain each conclusion from the premises. (c) Using a truth table, prove or disprove that the premises statements from part (a) and conclusion(s) from Part (b) form a tautology. (d) Prove or disprove the following the logical equivalence using the laws of logic: [c(b(ca))] and c(ab) (e) Write the negation of the statement (ii) and (iii) from part (a) in English. (f) Let Q(x,y) be the statement " x+y=xy. If the domain for both variables consists of all integers. Determine the truth value of the following statements. (i) yxQ(x,y) (i) yxQ(x;y) (g) Let C(x) be the statement " x is in the correct place" and E(x) be the statement " x is in excellent condition," and let T(x) be the statement " x is a tool", where the domain for the x consists of all tools. Use quantifiers to express each of these statements. (i) All tools are in the correct place and are in excellent condition. (ii) One of your tools is not in the correct place, but it is in excellent condition