Union and intersection of sets

Union of sets

Given two sets $A$ and $B$, the union of $A$ and $B$ it is $$A\cup B=\{x\in U|x\in Aorx\in B\}$$

The union of $A$ and $B$, is the set of elements $x$ of $U$ such that $x$ belongs to $A$, or $x$ belongs to $B$.

The union operation is associative, commutative and has an identity element:

• Commutative: $A\cup B=B\cup A$

• Associative: $(A\cup B)\cup C=A\cup (B\cup C)$

• Identity element: $A\cup \mathrm{\varnothing }=\mathrm{\varnothing }\cup A=A$

The union of two sets introduced above can be extended to multiple sets. Thus, the union of a finite number of sets is given by "successive unions": $$A_1\cup \dots \cup A_n=((A_1\cup A_2)\cup \dots )\cup A_n)$$

Because of the associative property, any order of "matches" to make the union leads to the same result. The union of sets can also be generalized to define the union of an infinite number of sets $A_k$. In this case, it is defined by: $${\cup }_k{A}_k=\{x\in U|\mathrm{\exists }k:x\in A_k$$

Intersection of sets

Given two sets $A$ and $B$, we define their intersection as $$A\cap B=\{x\in U|x\in Aandx\in B\}$$

The intersection of $A$ and $B$, is the set of elements $x$ of $U$, such that, $x$ belongs to $A$, and $x$ belongs to $B$.

The intersection operation is commutative, associative and it has identity and inverse element:

• Commutative: $A\cap B=B\cap A$

• Associative: $(A\cap B)\cap C=A\cap (B\cap C)$

• Identity element: $A\cap \mathrm{\varnothing }=\mathrm{\varnothing }\cap A=\mathrm{\varnothing }$

• Inverse element: $A\cap A^c=A^c\cap A=\mathrm{\varnothing }$, where $A^c$ represents the concept "complement".

Below we can see some properties that are satisfied between the intersections and unions.

$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

$(B\cup C)\cap A=(B\cap A)\cup (C\cap A)$ (distributive property regarding the union)

$A\cup (A\cap B)=A=A\cap (A\cup B)$ (absorption law)

The intersection of two sets can be extended to any number of sets. $$A_1\cap \dots \cap A_n=((A_1\cap A_2)\cap \dots )\cap A_n)$$

Due to the associative property, any order of "matches" to make the intersection leads to the same result. The sets of intersection can also be generalized to define the intersection of an infinite number of sets $A_k$. In this case it is defined as: $${\cap }_k{A}_k=\{x\in U|\mathrm{\forall }k:x\in A_k\}$$

Finally, two sets are called disjoint if their intersection is zero.