# Category:Indexed Families

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This category contains results about Indexed Families.

Definitions specific to this category can be found in Definitions/Indexed Families.

The image $\Img x$, consisting of the terms $\family {x_i}_{i \mathop \in I}$, along with the indexing function $x$ itself, is called a **family of elements of $S$ indexed by $I$**.

## Subcategories

This category has the following 6 subcategories, out of 6 total.

### D

- Disjoint Families of Sets (2 P)

### E

- Examples of Indexed Families (4 P)

### I

- Intersections of Families (3 P)

### S

### U

- Unions of Families (3 P)

## Pages in category "Indexed Families"

The following 45 pages are in this category, out of 45 total.

### C

### D

- De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection
- De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union
- De Morgan's Laws (Set Theory)/Set Difference/Family of Sets
- De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection
- De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Union

### I

- Image of Intersection under One-to-Many Relation/Family of Sets
- Image of Intersection under Relation/Family of Sets
- Image of Union under Relation/Family of Sets
- Indexed Cartesian Space is Set of all Mappings
- Intersection Distributes over Union/Family of Sets
- Intersection is Empty Implies Intersection of Subsets is Empty
- Intersection is Largest Subset/Family of Sets
- Intersection is Subset/Family of Sets
- Intersection of Family is Subset of Intersection of Subset of Family

### P

- Partition of Indexing Set induces Bijection on Family of Sets
- Partition of Indexing Set induces Bijection on Family of Sets/Lemma
- Preimage of Intersection under Mapping/Family of Sets
- Preimage of Union under Mapping/Family of Sets
- Preimage of Union under Relation/Family of Sets
- Projection from Product of Family is Surjective
- Projection is Surjection/Family of Sets

### S

- Set Intersection is Idempotent/Indexed Family
- Set Intersection is Self-Distributive over Family of Sets
- Set Intersection is Self-Distributive/Families of Sets
- Set Intersection Preserves Subsets/Families of Sets
- Set Intersection Preserves Subsets/Families of Sets/Corollary
- Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty
- Set is Subset of Union of Family
- Set is Subset of Union/Family of Sets
- Set of Sets can be Defined as Family
- Set Union is Self-Distributive/Families of Sets
- Set Union Preserves Subsets/Families of Sets

### U

- Union Distributes over Intersection/Family of Sets
- Union is Smallest Superset/Family of Sets
- Union of Indexed Family of Sets Equal to Union of Disjoint Sets
- Union of Inverses of Mappings is Inverse of Union of Mappings
- Union of Mappings which Agree is Mapping/Family of Mappings
- Union of Subset of Family is Subset of Union of Family