# Category:Cartesian Product

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This category contains results about cartesian products.

Definitions specific to this category can be found in Definitions/Cartesian Product.

Let $S$ and $T$ be sets or classes.

The **cartesian product** $S \times T$ of $S$ and $T$ is the set (or class) of ordered pairs $\tuple {x, y}$ with $x \in S$ and $y \in T$:

- $S \times T = \set {\tuple {x, y}: x \in S \land y \in T}$

## Subcategories

This category has the following 12 subcategories, out of 12 total.

### C

### E

### O

### P

## Pages in category "Cartesian Product"

The following 81 pages are in this category, out of 81 total.

### B

### C

- Canonical Injection into Cartesian Product of Modules
- Cardinality of Cartesian Product
- Cartesian Product Distributes over Intersection
- Cartesian Product Distributes over Set Difference
- Cartesian Product Distributes over Union
- Cartesian Product Exists and is Unique
- Cartesian Product is Anticommutative
- Cartesian Product is Anticommutative/Corollary
- Cartesian Product is Empty iff Factor is Empty
- Cartesian Product is Empty iff Factor is Empty/Family of Sets
- Cartesian Product is not Associative
- Cartesian Product is Set Product
- Cartesian Product is Set Product/Family of Sets
- Cartesian Product of Bijections is Bijection
- Cartesian Product of Bijections is Bijection/General Result
- Cartesian Product of Countable Sets is Countable
- Cartesian Product of Family is Empty iff Factor is Empty
- Cartesian Product of Family of Subsets
- Cartesian Product of Group Actions
- Cartesian Product of Homeomorphisms is Homeomorphism
- Cartesian Product of Intersections
- Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous
- Cartesian Product of Natural Numbers with Itself is Countable
- Cartesian Product of Semirings of Sets
- Cartesian Product of Sets is Set
- Cartesian Product of Subsets
- Cartesian Product of Subsets/Corollary 1
- Cartesian Product of Subsets/Corollary 2
- Cartesian Product of Subsets/Corollary 3
- Cartesian Product of Subsets/Family of Nonempty Subsets
- Cartesian Product of Subsets/Family of Subsets
- Cartesian Product of Unions
- Cartesian Product of Unions/Corollary
- Cartesian Product of Unions/General Result
- Cartesian Product with Complement
- Composition of Cartesian Products of Mappings
- Construction of Inverse Completion/Cartesian Product with Cancellable Elements
- Construction of Inverse Completion/Congruence Relation
- Construction of Inverse Completion/Equivalence Relation
- Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements
- Construction of Inverse Completion/Equivalence Relation/Members of Equivalence Classes
- Construction of Inverse Completion/Generator for Quotient Structure
- Construction of Inverse Completion/Identity of Quotient Structure
- Construction of Inverse Completion/Image of Quotient Mapping is Subsemigroup
- Construction of Inverse Completion/Invertible Elements in Quotient Structure
- Construction of Inverse Completion/Properties of Quotient Structure
- Construction of Inverse Completion/Quotient Mapping is Injective
- Construction of Inverse Completion/Quotient Mapping is Monomorphism
- Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism
- Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements
- Construction of Inverse Completion/Quotient Structure
- Construction of Inverse Completion/Quotient Structure is Commutative Semigroup
- Construction of Inverse Completion/Quotient Structure is Inverse Completion
- Correspondence between Set and Ordinate of Cartesian Product is Mapping
- Cross-Relation Equivalence Classes on Natural Numbers are Cancellable for Addition

### E

- Equality of Cartesian Products
- Equality of Ordered Pairs
- Equality of Ordered Tuples
- Equivalence Classes of Cross-Relation on Natural Numbers
- Equivalence of Definitions of Ordered Pair
- External Direct Product of Projection with Canonical Injection
- External Direct Product of Projection with Canonical Injection/General Result