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Statements, Connectives, Quantifiers

Overview. Section 3.1 in the textbook:StatementsConnectivesQuantifiers. Statements. . Statements. A statement (logic-wise) is a declarative sentence (i.e. one that is either true or false)Represented symbolically by using lowercase lettersConcerned only if it is possible for the sentence to evaluates to EITHER true OR falseStatements CANNOT be exclamations or questionsSee sentences e)

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Statements, Connectives, Quantifiers

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    1. Statements, Connectives, & Quantifiers MATH 102 Contemporary Math S. Rook

    2. Overview Section 3.1 in the textbook: Statements Connectives Quantifiers

    3. Statements

    4. Statements A statement (logic-wise) is a declarative sentence (i.e. one that is either true or false) Represented symbolically by using lowercase letters Concerned only if it is possible for the sentence to evaluates to EITHER true OR false Statements CANNOT be exclamations or questions See sentences e) – g) on pg 83 of the textbook What are examples of statements? There are hardly any wrong answers!

    5. Simple & Compound Statements A simple statement contains ONE idea Negating a statement, denoted as ~, means to write the opposite of the original statement e.g. Negate one of the statements we discussed previously A compound statement contains more than one idea by joining simple statements together using connectives (bridges) Four categories of connectives – we will discuss each in greater detail in a few slides: Conjunction & Disjunction Conditional & Biconditional

    6. Negation of Statements Ex 1: Write the negation of the statement symbolically: a) I forgot to feed the cat today. b) He does not cheat at cards.

    7. Connectives

    8. Conjunction & Disjunction A conjunction, symbolized by , is a compound statement that uses the word and to connect statements Why is a conjunction a compound statement? e.g. Take two additional statements from the “pool” and form a conjunction A disjunction, symbolized by , is a compound statement that uses the word or to connect statements e.g. Take two additional statements from the “pool” and form a disjunction

    9. Conjunction & Disjunction (Example) Ex 2: Consider the following statements. p: He can juggle. q: I know how to speak German. a) Express symbolically: He can not juggle and I know how to speak German. b) Write in English: ~p v ~q

    10. Conditional & Biconditional A conditional, symbolized by , is a compound statement that connects two statements in an if …, then structure e.g. Take two additional statements from the “pool” and form a conditional A biconditional, symbolized by , is a compound statement that connects two statements in an if and only if structure e.g. Take two additional statements from the “pool” and form a biconditional

    11. Conditional & Biconditional (Example) Ex 3: Consider the following statements. p: The defendant is convicted of perjury. q: He will spend at least 30 years in jail. a) Express symbolically: If the defendant is convicted of perjury, then he will spend less than 30 years in jail. b) Write in English:

    12. Quantifiers

    13. Quantifiers Quantifiers are words or phrases in a statement that answer the question “how many?” Two common classes of quantifiers: Universal: a phrase that indicates EVERY object satisfies a given property e.g. All, every e.g. Take a statement from the “pool” and modify it to use a universal quantifier Existential: a phrase that indicates ONE OR MORE objects satisfy a given property e.g. Some, there exists, there is at least one e.g. Take a statement from the “pool” and modify it to use an existential quantifier

    14. Negating a Universal Quantifier Consider negating the following statement: Every student in this room will get a ‘C.’ What does it mean for this statement to be false? How would we negate the statement? The negation of a universal quantifier is an existential quantifier

    15. Negating an Existential Quantifier Consider negating the following statement: Some students use a calculator. What does it mean for this statement to be false? How would we negate the statement? What is another, more concise, way to write this? The negation of an existential quantifier is a universal quantifier

    16. Negating Quantifiers (Example) Ex 4: Write the negation of each statement in English: a) All exams in this class require studying. b) Some professional wrestling matches are not scripted. c) At least one item on the McDonald’s menu is a healthy choice.

    17. Summary After studying these slides, you should know how to do the following: Identify statements and differentiate between simple and compound statements Negate a statement Write simple and compound statements symbolically Understand the meaning of quantifiers Negate statements that contain quantifiers Additional Practice: See the list of suggested problems for 3.1 Next Lesson: Truth Tables (Section 3.2)

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