ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Collection of sets
Definition D76
Set intersection
Formulation 0
Let $x$ be a D11: Set.
The intersection of $x$ is the D11: Set \begin{equation} \cap x : = \{ z \mid \forall \, y \in x : z \in y \} \end{equation}
Formulation 4
Let $X_j$ be a D11: Set for each $j \in J$.
The intersection of $\{ X_j \}_{j \in J}$ is the D11: Set \begin{equation} \bigcap_{j \in J} X_j : = \{ x \mid \forall \, j \in J : x \in X_j \} \end{equation}
Results
R4151: Binary intersection is a lower bound to each set in the intersection
R2220: Binary set intersection is commutative
R4149: Countable intersection is a lower bound to each set in the intersection
R4215: Countable set intersection is invariant under bijective shifting of indices
R4150: Finite intersection is a lower bound to each set in the intersection
R4216: Finite set intersection is invariant under bijective shifting of indices
R4148: Intersection is a lower bound to each set in the intersection
R1130: Intersection is largest lower bound
R2079: Isotonicity of set intersection
R4267: Set intersection with empty set is empty