Predicate

Abstract

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• Also known as Propositional Functions & Open Sentences
• Example: let P stand for is a student at NUS, then P(x) = x is a student at NUS, where x is the Variable, P is the Symbol
• Truth value depends on Variable
• Becomes a Mathematical Statement when specific values are substituted for Variable
• Returns either true or false, thus cant be used as a Variable that is substituted into Symbol like P()
• Made of Symbol & finite number of Variable

Terminologies

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Variable

• Holds value that determines if Predicate is true or false

true or false! Only Mathematical Statement can be true or false

Symbol

• Represents a property or relation

Quantifiers

• Instead of manually assign a specific value to Variable to obtain a Mathematical Statement
• Quantifiers is another way, specific how many of a particular type of values that make the predicate true

$$\forall$$ $$\exists$$

Order of Quantifiers

• Unless the Quantifiers are of the same type. Otherwise, the meaning is different

Example

1. For all people x, there is a person y such that x loves y
2. There is a person y such that all people x, x loves y

• The first one means for everyone (you, me, he), there is someone we love
• The second one means there is someone who is loved by everyone (you, me, he)

Domain of Predicate Variable

• The set of all values that may be substituted in place of the Variable
• Also known as Domain of Discourse, Universe of Discourse, Universal Set & Universe

Truth Set

• The set of values in Domain of Predicate Variable substituted to Variable that makes the Predicate true
• Detonated by the following expression, meaning value of x is an element of the Domain of Predicate Variable, such at the Predicate is true

$$x\in D\mid P(x)$$