Sol_Review Exam 1_sp17(4)

# Sol_Review Exam 1_sp17(4) - Mat 243 Exam 1 Review 1 Fill in...

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Mat 243 Exam 1 Review 1. Fill in the blank in the statements below: (a) Two propositions are logically equivalent if and only if they have the same truth values for all possible truth values of the variable. In other words, they share the same truth table. (b) A tautology is a proposition that is always truth for all possible truth values of the variables. (c) The negation of ”if p then q” is ? ∧ ¬? (d)” r is a necessary condition for s” means if …… s …… then …… r ……… (e) “p is sufficient condition for q means if …… p ……then … q ……. (f) “ r only if p ” means if … r …….then …… p …….. (2) Given the conditional: I go to the beach and I go out dancing in the evening, if I am done with my homework. (a) State the contrapositive of this statement: If I do not go to the beach or I do not go out dancing, then I am not done with my homework. (b) State the inverse of this statement: If I am not done with laundry, then I do not go to the beach or I do not go out dancing. (c) State the converse of this statement: If I go to the beach and I go our dancing in the evening, then I am done with my homework. (d) State the negation of this statement: I am done with my homework and I do not go to the beach or I do not go our dancing. (3) Write the following statements in symbolic form. Identify the propositional functions (if needed), universe of discourse you are using. Find the negation of each statement in symbolic form and in English. (a) For every real number there exists a larger real number. Domain for x and y: all real numbers: ∀?∃?(? > ?) . Negation: ∃?∀? (? ≤ ?) (b) Anna likes Dave but Dave does not like anyone. Domain: all people. Let L(x,y): x likes y. 𝐿(?𝑛𝑛𝑎, 𝐷𝑎𝑣𝑒) ∧ ∀?¬𝐿(𝐷𝑎𝑣𝑒, ?) Negation: ¬𝐿(?𝑛𝑛𝑎, 𝐷𝑎𝑣𝑒) ∨ ∃?𝐿(𝐷𝑎𝑣𝑒, ?) ≡ 𝐿(?𝑛𝑛𝑎, 𝐷𝑎𝑣𝑒) → ∃?𝐿(𝐷𝑎𝑣𝑒, ?) If Anna does not like Dave, then there is someone whom Dave likes.

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Mat 243 Exam 1 Review (c) everybody likes everybody. ∀?∀?𝐿(?, ?) Negation: ∃?∃?¬𝐿(?, ?) . Not everybody likes everybody. There is a person x and there is a person y such the x does not like y. (d) Everybody likes somebody. ∀?∃?𝐿(?, ?) Negation: ∃?∀?¬𝐿(?, ?). Not everybody likes somebody. Some people do not like anybody. (e) Somebody likes everybody. ∃?∀?𝐿(?, ?) Negation: ∀?∃?¬𝐿(?, ?) . Everybody does not like somebody. (f) Nobody likes everybody. ¬∃?∀?𝐿(?, ?) ≡ ∀?∃? ¬𝐿(?, ?) Negation: ∃?∀?𝐿(?, ?) . Somebody likes everybody. (g) Somebody likes nobody. (Somebody does not like anybody). This was the negation of (d).
• Spring '06
• Callahan
• Math, Logic, Rational number, Mohacsy

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