Truth Tables

MATH 102Contemporary 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– Is the statement true or false? What about its

negation?• 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

F T

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

qp qp

p q

T T T

T F F

F T F

F F F

qp

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 T T

T F T

F T T

F F 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)

rqp

rqp

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

qqp ~ ~

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)

qpqp ~ ~

rqp ~~

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

qpqp ~ ~ ~

qpqp ~ ~ ~

Logically Equivalent Statements (Example)

Ex 3: Determine whether the pairs of statements are logically equivalent:

a)b)

qpqp ; ~~~

rpqprqp ~ ~ ; ~ ~

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)