Union

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`Union`

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Union[list₁,list₂,…]

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

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Union[list]

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

Details and Options

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

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Give a sorted list of distinct elements:

In[1]:=1

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https://wolfram.com/xid/0giw0i-ruc

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0giw0i-hx3

Out[1]=1

Enter using un:

In[1]:=1

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https://wolfram.com/xid/0giw0i-vha

Out[1]=1

Scope (1)Survey of the scope of standard use cases

Give a list of the distinct lists:

In[1]:=1

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https://wolfram.com/xid/0giw0i-ewa1nk

Out[1]=1

Generalizations & Extensions (1)Generalized and extended use cases

Union works with any head, not just List:

In[1]:=1

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https://wolfram.com/xid/0giw0i-ds0

Out[1]=1

Options (4)Common values & functionality for each option

SameTest (4)

Use equivalence classes based on absolute value:

In[1]:=1

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https://wolfram.com/xid/0giw0i-gvm

Out[1]=1

Use equivalence classes based on Floor:

In[1]:=1

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https://wolfram.com/xid/0giw0i-bscbho

Out[1]=1

Use Total of list elements:

In[1]:=1

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https://wolfram.com/xid/0giw0i-hmxah

Out[1]=1

Use equality of last and first elements:

In[1]:=1

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https://wolfram.com/xid/0giw0i-dwkrbi

Out[1]=1

Applications (4)Sample problems that can be solved with this function

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

In[1]:=1

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https://wolfram.com/xid/0giw0i-lud

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0giw0i-qwm

Out[1]=1

Find the distinct elements in the iteration:

In[1]:=1

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https://wolfram.com/xid/0giw0i-ewu

Out[1]=1

Find what options are used by a list of functions:

In[1]:=1

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https://wolfram.com/xid/0giw0i-cnsc3w

Out[1]=1

Properties & Relations (2)Properties of the function, and connections to other functions

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

In[1]:=1

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https://wolfram.com/xid/0giw0i-p214s

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0giw0i-ckt3x8

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0giw0i-bu6xf7

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0giw0i-1pxsf

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0giw0i-cpcbue

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0giw0i-b0uwm7

Out[3]=3

Possible Issues (1)Common pitfalls and unexpected behavior

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

In[1]:=1

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https://wolfram.com/xid/0giw0i-b1g60q

In[2]:=2

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https://wolfram.com/xid/0giw0i-b4wxt1

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0giw0i-ir945y

Out[3]=3

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

In[4]:=4

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https://wolfram.com/xid/0giw0i-j7gq7r

Out[4]=4

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