Bridge to Sets, Relations, and Functions

Why This Bridge Exists

Symbolic logic gave us:

• symbols
• variables
• predicates
• propositions
• truth values
• inference rules
• quantifiers

But reasoning often needs structure around objects.

We need to talk about:

• collections of objects
• membership
• relationships between objects
• mappings from inputs to outputs

That is where sets, relations, and functions enter.

Sets, relations, and functions give structure to the objects logic talks about.

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From Domain To Set

Quantifiers depend on a domain.

Example:

For every x in the domain, P(x)

A set gives a precise way to describe that domain.

Example:

Domain = {1, 2, 3, 4}

Now a universal statement becomes clearer:

For every x in {1, 2, 3, 4}, P(x)

Sets make domains explicit.

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From Predicate To Relation

A predicate can describe a property.

Example:

Even(x)

A predicate can also describe a relationship.

Example:

GreaterThan(x, y)

Relations give a formal way to study those connections.

Example:

GreaterThan = {(5, 3), (4, 2), (9, 1)}

Relations make predicates over multiple objects structural.

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From Function Symbol To Function

Logic introduced function symbols.

Example:

f(x)

Sets and functions make this more precise:

function = mapping from a domain to a codomain

Example:

f(x) = x + 1

Mapping:

1 -> 2
2 -> 3
3 -> 4

Functions formalize rule-governed mappings.

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Why This Matters For Computation

Computation also depends on structure.

Example:

input -> function -> output

or:

state -> transition relation -> next state

Sets describe possible states.

Relations describe allowed transitions.

Functions describe deterministic mappings.

Sets, relations, and functions connect logic back to computation.

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What Comes Next

The next chapter builds these pieces bottom-up:

• element
• set
• membership
• subset
• relation
• ordered pair
• function
• domain
• codomain
• range
• composition
• inverse

The goal is to make structure explicit.

Once structure is explicit, we can reason about graphs, transformations, vectors, and eventually models.

The next step is understanding structured objects and mappings.

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Final Summary

• Logic needs domains, objects, and relationships.
• Sets define collections and domains.
• Relations describe connections between objects.
• Functions describe mappings from input to output.
• Predicates can be understood structurally through relations.
• Function symbols become precise through mathematical functions.
• Sets, relations, and functions bridge symbolic logic back to computation and forward to graph structure.