Transcript p ^ q - IB Math

Logic
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.
Negation and Venn
Diagrams
Venn Diagrams can be use to represent a proposition and
its negation.
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.
The truth table for conjunction p ^ q
p
q
T
T
T
F
F
F
T
F
F
F
F
p^ q
T
In a Venn diagram representing two propositions,
the intersection of the Venn diagram represents
the conjunction proposition.
U
p^q
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
pVq
pVq
Truth Table for p V q and p V q
p
q
pVq
p
q
pVq
T
T
T
T
T
F
T
F
T
T
F
T
F
T
T
F
T
T
F
F
F
F
F
F
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 → q
p is called the antecedent and q is
called the consequant.
The truth of the implication (→) 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 → q and q → p.
That is p ↔ q = (p → q) ^ (q → p).
Converse, Inverse and
Contrapositive
The converse of the statement p → q
is the statement q → q.
The inverse of the statement p → q is
the statement ¬ p → ¬ q.
The contrapositive of the statement p
→ 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’.
Syllogism
A syllogism is an argument consisting
of three propositions. The first two
of which are true and the third is
supposed to be the logical conclusion
of the first two.
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