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

This preview shows page 1 - 3 out of 7 pages.

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.
Image of page 1

Subscribe to view the full document.

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).
Image of page 2
Image of page 3
  • Spring '06
  • Callahan
  • Math, Logic, Rational number, Mohacsy

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern