Geometry Lesson Notes 2.2 Date ______
Objective: Determine truth values of statements, negations, conjunctions, and disjunctions.
Construct truth tables.
Statement: any sentence that is either true or false, but not both.
symbol: a statement can be represented by a letter, e.g. p or q
Truth value: the truth or falsity of a statement
Negation of a statement: has the opposite meaning and the opposite truth value of the
statement.
symbol: the negation of a statement, p, is represented by ~p, read not p.
p: Merrimack is the capital of NH false
~p: Merrimack is not the capital of NH true
Compound statement: formed by joining two or more statements.
Compound statements can be formed by joining the statements with and or or.
Conjunction: a compound statement formed by joining two or more statements with
the word and.
symbol: p Ù q, read p and q
A conjunction is true ONLY when ALL statements are true!
Disjunction: a compound statement formed by joining two or more statements with
the word or.
symbol: p Ú q, read p or q
A disjunction is true if at least one statement is true!
(could be one, some, or all statements are true.)
Example 1 & 2 (pp 68-69): Truth Values of Conjuctions / Disjunctions
Find the truth value of each statement below:
Concord is a city in NH and Concord is the capital of NH.
Concord is a city in NH or Concord is the capital of NH.
Merrimack is a city in NH and Merrimack is the capital of NH.
Merrimack is a city in NH or Merrimack is the capital of NH.
Tampa is a city in NH and Merrimack is the capital of NH.
Tampa is a city in NH or Merrimack is the capital of NH.
Tampa is a city in NH and Concord is the capital of NH.
Tampa is a city in NH or Concord is the capital of NH.
Venn Diagrams: sets or parts of the whole are represented by circles.
Overlapping areas represent members that are part of two (or more) sets.
Overlapping areas are called the intersection of two or more sets and indicate members
that satisfy a conjunction.
Sum of all circles is called the union of two or more sets and indicates members that
satisfy a disjunction.
Example 3 (p 70): Use Venn Diagrams.
Students enrolled at a Dance School
How many students are enrolled in tap or jazz?
Tap and jazz?
Just ballet?
Make up your own statements.
Truth Table: a method for organizing truth values.
Use truth values for negation, conjunction, and disjunction to construct truth tables for
more complex compound statements.
Be sure to list all
combinations of truth
values for the statements.
(Use Fundamental Counting
Principle to determine
number of rows.)
¯ ¯
Negation
p ~ p
T F ¬ If p is true, then ~p is false.
F T ¬ If p is false, then ~p is true.
Conjunction
p q pÙ q
T T T ¬ A conjunction is true only when both statements are true.
T F F
F T F
F F F
Disjunction
p q pÙ q
T T T
T F T
F T T
F F F ¬ A disjunction is false only when both statements are false.
Example 4 (p 71): Construct Truth Tables
Construct a truth table for each compound statement.
a. ~ p Ú q two statements with two possible truth values: 2 ´ 2 = 4 rows
p q ~ p ~ p Ú q 1st statement 2 values 2nd statement 2 values
______
______
______
______
b. p Ú (~ q Ù r) three statements with two possible truth values: 2 ´ 2 ´ 2 = 8 rows
p q r ~ q ~ q Ù r p Ú (~ q Ù r)
____ ______
______
______
______
______
______
______
______
Evaluate the compound statement given:
p: Concord is a city in NH.
q: Concord is the capital of NH.
r: Tampa is a city in NH.
c. ( p Ú q) Ù ~ r three statements with two possible truth values: 2 ´ 2 ´ 2 = 8 rows
p q r ~ r p Ú q ( p Ú q) Ù ~ r
______
______
______
______
______
______
______
______
Evaluate the compound statement given:
p: Concord is a city in NH.
q: Merrimack is the capital of NH.
r: Tampa is a city in NH.
HW: A2a pp 72-74 #18-44 even, 45-47, 54
A2b pp 72-74 #19-43 odd, 51-52
A2c 2-2 Skills Practice / Practice
fms-Geometry Lesson Notes 2.2 Page 1 of 5