1.PROPOSITIONAL LOGIC
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| Material Equivalence - Asserts the statements always have the same truth value
2.PROPORTIONALEQUIVALENCES
Definition: two
propositional form on the same variables (logically) equivalent if they have
the same result column in their truth table notation
.
—Tautology =
proposition that is always true
—Contradiction =
proposition that is always false
—Contingency =
proposition that is neither tautology nor contradiction
 
Contigency
Two compound proposition p and q are logically
equivalent if the columns in a truth table giving their truth values agree.
p
≡q if and only if p q is a tautology.
3.PREDICATES AND QUANTIFIERS
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Proposition
a possible condition of the world about which want to say
something.
Propositional Variables
a variable which can be the true or false.
Types of Truth Table
Negation
Conjunction - assert both statement are true
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| Disjunction - Asserts at least one statement is true |
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| Material implication - assume it is true unless proven false |
Introduction to Predicates
➳ Also known as propositional function
➳ Sentences that contain variable either true or false
depending on value assign to variables.
➳Denote by P(x)
Exp : P(x): x>3
Definition Quantifiers
➳Quantifers are words that refer to quantities such as
"some" or "all" and tell for
how
many elements a given predicate is true
Two type
of quantifiers
universal quantifier:
DEFINITION
v predicate is true for all values
v symbol: "
v read: for all
example
P(x) : “x must take a discrete mathematics course”
Q(x): “x is a computer science student”.
Express the statement “Every computer science student
must take a discrete mathematics course”.
"x(Q(x) → P(x))
Express the statement “Everybody must take a discrete
mathematics course or be a computer science student”.
"x(Q(x) ˅ P(x))
existential quantifier:
DEFINITION
v predicate is true for some values
v symbol: $
v read: there exists
P(x) : “x must take a discrete mathematics course”
Q(x): “x is a computer science student”.
Express the statement “some computer science student
must take a discrete mathematics course”.
$x(Q(x) → P(x))
Express the statement “some student must take a discrete
mathematics course or be a computer science student”.
$x(Q(x) ˄ P(x))
EXAMPLES OF USING QUANTIFIERS IN REALITY
Assume:
A(x) : x is a apple.
B(x) : x is a banana.
C(x) : x is cherry.
and the universe of discourse for all three functions is the set of all fruits.
- Everything is a apple : "x A(x)
- All apple are banana : "x [ A(x) → B(x) ]
-Some are banana : $x B(x)
