Download Chapter 15 Logic Name Date Objective: Students will use

Chapter 15 Logic
Name _______________________________________________ Date __________________
Objective: Students will use propositions to create truth tables and logical equivalences in order
to draw logical conclusions
Proposition
Propositions are statements that may be true or false.
Propositions may be indeterminate - a proposition that is not certain.
Notations used for propositions
Letters such as p, q, and r are used to represent propositions.
Negation
The negation of a proposition is its negative.
The negation of proposition p is written as ¬p.
Truth values are T for true and F for false.
Proportion p
p
Negative of Proportion p
¬p
T
F
F
T
The wording of a negation may depend on the domain of the variable.
Negation and Venn Diagrams
Venn Diagrams can be use to represent a proposition and its negation.
U
P
P’
If p is a proposition and ¬p is its negation, they can be represented as shown below.
U
p
¬p
Compound Propositions
Compound Propositions are statements which are formed using connectives such as and or or.
Conjunction
When two propositions are joined by the word and, the new proposition is the conjunction of the
original propositions.
If p and q are propositions, then p ^ q stand for their conjunction p and q
The truth table for conjunction p ^ q
p
q
p^q
In a Venn diagram representing two propositions, the intersection of
the Venn diagram represents the
T
T
T
T
F
F
F
T
F
F
F
F
conjunction proposition.
Disjunction
When two propositions are joined
by the word or, the new proposition is the
disjunction of the original propositions.
If p and q are propositions, then p V q stands for their inclusive disjunction and p V q
stand for their exclusive disjunction.
The inclusive disjunction is true when one or both propositions are true, since in this case p or q
means p or q, or both p and q.
i.e. p V q = p or q or both p and q
The exclusive disjunction is true when only one of the propositions is true, since in this
case p or q means p or q but not both.
i.e. p V q = p or q but not both
Venn Diagram for p V q and p V q
Truth Table for p V q and p V q
Truth Table and Logical Equivalence
If two compound propositions have the same T/F column they are said to have logical
equivalence (logically the same).
Example: ¬ (p ^ q) and ¬p V ¬q are logically equivalent.
Truth table for ¬ (p ^ q) is:
p
q
p^q
¬(p ^ q)
T
T
T
F
T
F
F
T
F
T
F
T
F
F
T
T
The truth table for ¬p V ¬q is:
p
q
¬p
¬q
¬p V ¬q
T
T
F
F
F
T
F
F
T
T
F
T
T
F
T
F
F
T
T
T
The result ¬ (p ^ q) = ¬p V ¬q and ¬ (p V q) = ¬p V ¬q are called deMorgan properties.
Tautologies
A tautology is a compound statement which is true for all possibilities in the truth table.
A logical contradiction is a compound statement which is false for all possibilities in the truth
table.
Truth Tables for Three Propositions
When three propositions are combined in a truth table there will be 8 possibilities.
p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
Implication
•
If a compound statement can be formed using an “if …, then ….” means of connection
then the statement is an implication.
•
The statement is called an implicative statement.
•
Using symbol, the statement “if p …, then q …” as p
•
p is called the antecedent and q is called the consequant.
•
The truth of the implication (
q
) is only false when p is true and q is false.
•
That is; when the antecedent is true and the consequant is false.
Equivalence
Two statements are equivalent if one implies the other or vise versa.
Equivalence is denoted by the symbol
.
For two statements p and q, p
q is the conjunction of the two implications p
That is p
p).
q = (p
q) ^ (q
q and q
Converse, Inverse and Contrapositive
The converse of the statement p
The inverse of the statement p
q is the statement q
q is the statement ¬ p
The contrapositive of the statement p
q.
¬ q.
q is the statement ¬ q
¬ p.
Valid Argument
An Argument is made up of premises (propositions) that lead to a conclusion.
The conclusion is usually indicated by the words “therefore” or “hence’.
Important Points
•
p ᴧ q is only T when p is T and q is F
•
p ᴠ q is F only when p is F and q is F
•
p ⟹ q is F only if p is T and q is F; that is implicative statement is F only when the
antecedent is T and the consequent is F because TF is not possible
•
p⟺q=p⟹qᴧq⟹p
•
the implication and its contra positive are logically equivalent
•
the converse and its inverse of an implication are logically equivalent
p.