# Union

Union[list1,list2,]

gives a sorted list of all the distinct elements that appear in any of the listi.

Union[list]

gives a sorted version of a list, in which all duplicated elements have been dropped.

# Details and Options

• If the listi are considered as sets, Union gives their union.
• Union[list1,list2,] can be input in StandardForm and InputForm as list1list2. The character can be entered as un or \[Union].
• The listi must have the same head, but it need not be List.
• Union[list1,,SameTest->test] applies test to each pair of elements in the listi to determine whether they should be considered the same.

# Examples

open allclose all

## Basic Examples(3)

Give a sorted list of distinct elements:

Give a sorted list of distinct elements from all the lists:

Enter using un:

## Scope(1)

Give a list of the distinct lists:

## Generalizations & Extensions(1)

Union works with any head, not just List:

## Options(4)

### SameTest(4)

Use equivalence classes based on absolute value:

Use equivalence classes based on Floor:

Use Total of list elements:

Use equality of last and first elements:

## Applications(4)

Find divisors that occur in any of 10, 12, and 20:

Find all the triples of bits that occur in the binary decomposition of 10!:

Find the distinct elements in the iteration:

Find what options are used by a list of functions:

## Properties & Relations(2)

Split on the sorted set gives lists of the same elements:

The union is equivalent to the first elements of these lists:

Tally gets the count of identical elements and returns them in the original order:

The union is the sorted list of the elements returned by Tally:

## Possible Issues(1)

For large sets Union may be slow with SameTest since it requires all pairwise comparisons:

When equivalence class representatives can be found, it may be faster to use Union on these:

The results are the same except for the choice of representative:

Wolfram Research (1988), Union, Wolfram Language function, https://reference.wolfram.com/language/ref/Union.html (updated 1996).

#### Text

Wolfram Research (1988), Union, Wolfram Language function, https://reference.wolfram.com/language/ref/Union.html (updated 1996).

#### CMS

Wolfram Language. 1988. "Union." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/Union.html.

#### APA

Wolfram Language. (1988). Union. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Union.html

#### BibTeX

@misc{reference.wolfram_2024_union, author="Wolfram Research", title="{Union}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Union.html}", note=[Accessed: 17-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_union, organization={Wolfram Research}, title={Union}, year={1996}, url={https://reference.wolfram.com/language/ref/Union.html}, note=[Accessed: 17-July-2024 ]}