Predicate

Abstract

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• Also known as Propositional Functions & Open Sentences
• Predicate is a function(Predicate Symbol) that accepts a valid input to its Predicate Variable and return either true or false
• Example: let $P()$, the predicate symbol stands for is a student at NUS and $x$ be the predicate variable. Together, we get $P(x)$ which stands for $x$ is a student at NUS

Dynamic Truth Value

Depends on the values we substituted to the Predicate Variable

Not a Mathematical Statement

Only becomes a Mathematical Statement when specific values are substituted to the Predicate Variable, and we call it Closed Predicate. Otherwise Open Predicate

Predicate is not a Predicate Variable

Can’t be used as a Predicate Variable that is substituted into Predicate Symbol like $P()$. Because Predicate is meant to return either true or false with value substituted

Predicate Variable

• Holds value that determines if Predicate is true or false

Not a Mathematical Statement

It doesn’t hold the value of true or false! Only Mathematical Statement can be true or false

Predicate Symbol

• Represents a property or Relation

Domain of Predicate Variable

• The set of all values that may be substituted in place of the Predicate 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 Predicate Variable that makes the Predicate return true
• Detonated by $x\in D\mid P(x)$, meaning a particular value $x$ is an element of the Domain of Predicate Variable, such that the Closed Predicate is true

Quantifiers

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• Instead of manually assign a specific value to Predicate Variable to obtain a Mathematical Statement
• Quantifiers is another way to specify how many of a particular type of values that can make the Predicate a closed predicate
• There are 2 types, represented with either $\forall$ or $\exists$

Order of Quantifiers

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

Given $P(x,y)$ is x loves y

1. $\forall x\exists y,P(x,y)$ in english: For all people x, there is a person y such that x loves y.

2. $\exists y\forall x,P(x,y)$ in english: 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)

Multiply-Quantified

Predicate with more than one Quantifiers

Implicitly Quantified

The Quantifiers are assumed without specified explicitly