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Propositional, Predicate Logics and Set

Getinet Yilma  Propositional logic  Predicate Logic  Set Theory

Propositional logic

 Introduction to Propositional logic  Proposition Operators  Proposition Operators and Truth Table  Result terminology

Propositional Logic…

• A proposition is a statement that can be either true or false; it must be one

or the other, and it cannot be both.

• The following are propositions:
  • the reactor is on;
  • the wing-flaps are up; and
  • a program P terminates.
• The following are not propositions
  • are you going out somewhere?
  • and 2+
• Propositions can be atomic (single truth value) or complex (multiple joined

with connectives)

Proposition connectives

• Compound propositions combined simple ones with Logical operators
  • A truth table for some formula p, q and operators describe the behavior of a formula under all possible interpretations.

p q p  q p  q p  q p  q

T T T T T T

T F F T F F

F T F T T F

F F F F T T

P  P

T F

F T

Example

• Nebil and Yasin are cousins if their fathers are brothers
  • P: Nebil and Yasin are Cousins
  • Q: Their fathers are brothers
  • P=>Q A polygon is a triangle if and only if A polygon has exactly 3 sides
  • P: A polygon is a triangle.
  • Q: A polygon has exactly 3 sides
  • P: x + 2 = 7 Q: x = 5
  • PQ
• Tautology : A compound proposition which is always true P or ~P is always true
• Contradiction : A compound proposition which is always false
• Satisfiable (consistent) happen when at least one (1) truth value brings the TRUE result.

Examples

• Logically Equivalent If PQ is always true then P and Q are said to be logically

equivalent.

• We write P≡Q e.g. ~(𝑃 ˅𝑄) ≡ ~𝑃˄~Q

Verify the following logical equivalences

1. p(q(  q)) ≡ p
2. (p (  q)) ≡ (  p)  q Verify truth tables to check whether the following are tautologies, contradictions or neither. p  (p  q) q  (p ( p)) p  (p  q)

• Can an atomic statement be a tautology or a contradiction? Explain

Propositional logic: limitations

• Propositional logic: the world is described in terms of elementary propositions and their logical combinations. - A program P terminates - Program refers to object, terminates refers properties
• Objects and properties are hidden in the statement, it is not possible to reason about them
• Statements that hold for many objects must be enumerated
• Example: –
• John is a SE Student  John has passed SEng
• Ann is a SE Student  Ann has passed SEng
• Ken is a SE Student  Ken has passed SEng
  • Replacing John, Ann and Ken by a variable x
  • x is a SE Student  x has passed SEng

Propositional logic: limitations

• In Propositional logic statements that define the property of the group of objects are not quantified. - All new cars must be registered. - Some of the light switches can be turned off.
• We can use quantifiers
• Universal quantifier – the property is satisfied by all members of the group  new cars must be registered
• Existential quantifier – at least one member of the group satisfy the property  the light switches can be turned off

Predicate logic

 Introduction predicate logic  Quantifiers  Application  Correctness

Predicate Logic

• Predicate logic , first-order logic or quantified logic is a formal language in

which propositions are expressed in terms of predicates, variables and

quantifiers.

• Quantifiers express the extent to which a predicate is true over a range of

elements.

• For example, imagine we have the statement: “Every person who is 21 years

of age or older is able to purchase alcohol. Sarah is 21 years old.”

• what does “every person” really mean?
• As the range of possibilities is too broad, we need to apply one of two types of

quantifiers to help us achieve our goal:

1. Universal Quantifiers
2. Existential Quantifier

Predicate Logic

• Existential Quantifier is the proposition that a property is true for some value

in a particular domain.

• There exists an x such that P(x)
• There exists an element x in the domain such that P(x)
• For some x, P(x)
• There is some x such that P(x) And is symbolically denoted Suppose we have two predicate variables x and y, where the domain for x is F = {foxes} and y has the domain S = {snails}, where P(x,y) is “Foxes are faster than Snails.”  x  D, P(x) “there exists x, belonging to domain D, such that p(x) is true

Predicate Logic

Quantifiers In Truth Table

Predicate Logic

Quantifier Negation

• Using the five circles, as seen above, we make the following statement: “Some circles are shaded.”
• What would the negation of this statement be?
• “Some circles are not shaded.” is correct?
• No! Because this statement is also trues
• Remember, a negation must have the opposite truth value.
• “All circles are not shaded.” is correct? Yes.
• The trick is to flip the universal quantifier to an existential quantifier or vice versa, and then negate the statement, “Some circles are shaded.”  “ All circles are not shaded .”
• Quantifiers give the ‘range’ and ‘universe’ over which the statement is being claimed.

Predicate Logic

The negation of quantification has the following properties: