Statements, Connectives, & Quantifiers

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 • StatementsCANNOT 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)