PROPOSITIONAL LOGIC Proposition or statements, are also known as simple sentences makes up the basic units for propositional logic. When two or more simple sentences are related through connectives, it forms a more complicated proposition called compound sentence. A statement / propositions must always be able to be validated as True (T) or False (F).
EXAMPLE:i) All fathers are men. This statement is true because only man can be father.ii) All men are father. This statement is false because only those men who are married and have kids are fathers. Single men are not father.iii) I like Monday. This sentence is neither true nor false as it does not include a fact. The notion of ‘liking Monday’ is entirely depending on the particular individual. Some may like it. Some may not. Sentence which are neither true nor false are not propositions.
PROPOSITIONAL VARIABLES Propositional variables are used to stand for unspecified statements. Means that each variableassigned may or may not have connection between them.
EXAMPLE: i) She likes dancing. ii) She likes camping. Iii) She likes outdoor activities.Each variable is assigned to a statement.
PROSITIONAL EQUIVALENCETautology• A compound proposition that is always trueContradiction• A compound proposition that is always falseContingency• A compound proposition that is neither a tautology nor a contradiction.
EXAMPLE:LOGICAL EQUIVALENCES• Compound proposition that have the same logical content. Means, compound propositions p and q are called logically equivalent if p ↔ q is a tautology.
PREDICATESIn some situations, a statement can be composed oftwo parts, the variable and the predicates.
EXAMPLE: The cost of an apple is RM x. (In this case, the cost of the apple is a predicate, while x is the variable which represents the price.) The above statement can also be expressed as propositional function P at the value of x, which is P(x), Where P[the cost of an apple] is affected by x[the price]. Note: Predicate is not a proposition until variable is declared on it. P itself is not a proposition until it is bound by x, forming P(x).
QUADTIFIERSUniversal Quantifier• ∀ is used to represent ‘for all’Existential Quantifier• ∃ is used to represent ‘for some state / value of’Uniqueness Quantifier• ∃! Is used to represent ‘for one and only one x’