MATH 102 Contemporary Math S. Rook Truth Tables MATH 102 Contemporary Math S. Rook

Overview Section 3.2 in the textbook: Listing all truth possibilities Truth tables for negation & connectives Evaluating compound statements Logically equivalent statements

Listing All Truth Possibilities

Listing All Truth Possibilities Consider a statement p What are the only possible truth values of p? Consider statements p and q We know the possible truth values of p and q separately What are all the possible truth values if we pick a value for p and then pick a value for q? What are all the possible truth values for statements p, q, and r? Notice the pattern?

Listing All Truth Possibilities (Continued) There will be 2k total possibilities for k statements Need to account for all possibilities Use the half-and-half method to capture all possibilities Populating the truth table with all possible truth values of the variables is very important to master! You must be able to work with a truth table containing up to 3 variables

Truth Tables for Negation & Connectives

Truth Tables Truth table: systematic way of determining the truth values of a compound statement by examining all the possible truth values of its input statements We looked extensively on populating a truth table with input values The goal for now is to be able to construct a truth table to show the possible truth values for ANY compound statement Later, we will see how to use truth tables to make logical inferences Need to first understand the truth tables for negation and connectives

Truth Table for Negation Consider the statement 2 + 1 = 3 Is the statement true or false? What about its negation? Consider the statement 8 – 5 = 2 Negation: If a statement is true, then its negation is false; if a statement is false, then its negation is true p ~p T F

Truth Table for Conjunction Let p be the statement I need to buy bread and q be the statement I need to buy milk What is the statement in English? What does it mean for to be true? Conjunction: True when ALL variables are true; false otherwise p q T F

Truth Table for Disjunction Let p be the statement I washed the cat and q be the statement I put my shoes on What is the statement p v q in English? What does it mean for p v q to be true? Disjunction: True when at least one variable is true; false when all variables are false p q p v q T F

Inclusive Or vs Exclusive Or The disjunction defined on the last slide is the inclusive or Version used in logic Different from the normally used exclusive or i.e. One or the other, but NOT BOTH e.g. If a waitress asks you whether you want Coke or Sprite, what does she expect you to say? When you see the word or in this chapter, we are referring to the inclusive or

Truth Tables (Example) Ex 1: Create a truth table for the following statements: a) b)

Evaluating Compound Statements

Evaluating Compound Statements Consider evaluating the compound statement How many variables are there? We know how to construct all possible inputs for a truth table Recall the Order of Operations in Algebra Similar construct in logic: Parentheses Negation Conjunction & Disjunction

Evaluating Compound Statements (Continued) Evaluate one negation or one connective per column What should we evaluate first in the example? Take ONE STEP at a time and keep adding columns to the truth table until you arrive at the desired statement Reduces the number of columns under consideration in each step to 2 or even 1 which is much easier! Requires practice in order to master!

Evaluating Compound Statements (Example) Ex 2: Construct a truth table: a) b)

Logically Equivalent Statements

Logically Equivalent Statements We say that two statements are logically equivalent if their truth values match exactly Useful to test whether two statements logically mean the same thing Use a truth table DeMorgan’s Laws deal with distributing a negation through parentheses to create a logically equivalent statement

Logically Equivalent Statements (Example) Ex 3: Determine whether the pairs of statements are logically equivalent: a) b)

Summary After studying these slides, you should know how to do the following: Populate a truth table with all combinations of truth values of the inputs Know the truth tables for negation, conjunction, and disjunction Evaluate a compound statement using a truth table Determine whether pairs of statements are logically equivalent Additional Practice: See the list of suggested problems for 3.2 Next Lesson: The Conditional & Biconditional (Section 3.3)