2. Translate the following English sentences into propositional formulas. Remember to define your atomic propositions! ... We Alle W wided that such intrini (1) The absence of antibodies in the subject's body implies that they are not susceptible to infection. (g) Participants were not timed on this task; however, most finished in less than 8 minutes. (h) The printer can hold a black ink cartridge or a color cartridge, but not both. () Only if a person has a real desire to want to change and puts forth real effort to make those changes do they come about. (Hint: You'll need three separate atomic propositions to fully translate this one.) 6) The existence of a Lyapunov function is a necessary and sufficient condition for the stability of the system. (k) Passing the Turing Test is a sufficient condition for a being's intelligence. (1) Disciplinary knowledge and organizational skills are necessary conditions for effec- tive teaching. (Hint: You should have three separate atomic propositions here.) (m) Bonus: Unless some dude is standing in the same spot for more than half of the pictures, the algorithm will select the correct background color. 3. Let I, M, and E have the following meanings: 1: The command returns an integer. M: The command displays a message. E: The command throws an exception. Express each of the following propositions as an English sentence. (a) IVE (b) IM (C) MI (d) - 1 (e) EAM (f) IAM (8) EV (IAM) 4. Bonus: Find an English sentence of the form p, and q." that means the same thing as the sentence "if p, then q." To be clear, you are not allowed to add any extra stuff to make things work out, so if you're trying to include any of the standard implication words like "if," "when," "case, "provided." "necessary," or "sufficient," then you've missed the point. Hint 1: If you're thinking about using words like "90" "hen," "hus," "herefore," "hence," "because," "since," or anything else that has to do with cause and effect, then you're on the wrong track entirely. Hint 2: Remember that conditionals have nothing to do with cause and effect. So for example, the sentence "It rained, and my car got wet," is not a valid answer, because it does not mean the same thing as "If it rained, then my car got wet." This is a pretty tough problem. The vast majority of "answers" I get do not fit the requirements for the problem. If you're in doubt about your example, you can send it to me ahead of time, and I'll take a look at it. **