• Introduction
• Iteration
  • Summation
  • Product
  • Set Notation
  • Nested Iteration
  • Conditions on indices
  • Union and Intersection
  • General Iteration
• General Symbols
  • Cases
  • Function Composition
  • Set Containment
  • Kronecker
• Operations
  • Factorial
  • Binomial
• Differentiation
  • Derivative
  • Partial Derivative
  • Gradient

Union $\bigcup$, intersection $\bigcap$

The union $\bigcup$ is the repeated application of the set union operation $\cup$. A set is just a "collection of unique things", so no duplicates. Most programming languages have some kind of set type included. The union of two sets is a new set that contains all entries of both sets. You can think of $\cup$ as a cup that gathers things falling into it.

The intersection $\bigcap$ is the repeated application of the set intersection operation $\cap$. The intersection of two sets is a new set containing only those elements that are found in both sets. Elements only in one of the sets are not included.

Basic forms: $\bigcup\limits_{i=0}^n f_i$ or $\bigcup_{i=0}^n f_i$ and $\bigcap\limits_{i=0}^n f_i$ or $\bigcap_{i=0}^n f_i$

Parts: See the section about Sum or Product

Meaning: Apply a union or intersection with each term $f_i$, starting at $i=0$ and ending at $i=n$. If the upper stop value is lower than the initial value, then the union is the empty set $\emptyset$, e.g. $\bigcup\limits_{i=1}^0 = \emptyset$. The intersection is a bit more tricky and you probably won't encounter this issue when implementing an algorithm. The result of the empty intersection is the set of all possible elements. So, if we call that set $A$ (for "all", sometimes also $\Omega$ ("Omega")), then we have $\bigcap\limits_{i=1}^0 = A$.

For the union $\bigcup$ this results in:

For the intersection $\bigcap$ this results in: