Function

Definition

A function maps input values to output values.

Example:

f(x) = x + 1

If:

x = 5

then:

f(5) = 6

Function = rule-governed mapping from input to output.

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Function Name And Input

In:

f(x)

f is the function name.

x is the input placeholder.

The expression says:

apply function f to input x

If:

f(x) = x + 1

then f names the rule:

add 1

A function symbol names a rule for producing output.

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Function vs Value

A function is not one value.

It is a mapping that can produce values.

Example:

f(x) = x + 1

Outputs:

f(1) = 2
f(2) = 3
f(3) = 4

The function stays the same.

The output changes with the input.

A function is a rule; its outputs depend on inputs.

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Function In Logic

In logic, a function can build a new term from an existing term.

Example:

father(x)

This can mean:

the father of x

If:

x = Ada

then:

father(Ada)

refers to a specific object under the interpretation.

Logical functions create terms from other terms.

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Function vs Predicate

A function returns a value or object.

Example:

double(3) = 6

A predicate returns a truth value.

Example:

Even(3) = false

This distinction matters because functions produce terms, while predicates produce statements that can be true or false.

Function returns an object or value; predicate returns true or false.

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Final Summary

• A function maps inputs to outputs.
• A function name represents a rule.
• The same function can produce different outputs for different inputs.
• In logic, functions can build terms.
• A function is different from a predicate.
• Functions help connect symbolic structure to rule-governed transformation.