Editing Relational algebra (section)

=== Antijoin ===

The antijoin (▷), written as ''R'' ▷ ''S'' where ''R'' and ''S'' are [[relation (database)|relation]]s,{{efn|In [[Unicode]], the Antijoin symbol is ▷ (U+25B7).}} is similar to the semijoin, but the result of an antijoin is only those tuples in ''R'' for which there is ''no'' tuple in ''S'' that is equal on their common attribute names.<ref name=unnesting>{{Cite conference |title=Unnesting Arbitrary Queries |conference=BTW |last=Neumann |first=Thomas |url=https://dl.gi.de/handle/20.500.12116/2418 |year=2015 |doi=}}</ref>

For an example consider the tables ''Employee'' and ''Dept'' and their
antijoin:

{{col-begin|width=auto; margin:0.5em auto}}
{{col-break}}
{| class="wikitable"
 |+           ''Employee''
 |-
 ! Name   !! EmpId !! DeptName
 |-
 | Harry  || 3415  || Finance
 |-
 | Sally  || 2241  || Sales
 |-
 | George || 3401  || Finance
 |-
 | Harriet || 2202 || Production
|}
{{col-break|gap=2em}}
{| class="wikitable"
 |+           ''Dept''
 |-
 ! DeptName   !! Manager
 |-
 | Sales  || Sally
 |-
 | Production || Harriet
|}
{{col-break|gap=2em}}
{| class="wikitable"
 |+  ''Employee''  ▷ ''Dept''
 |-
 ! Name   !! EmpId !!  DeptName
 |-
 | Harry  || 3415  || Finance
 |-
 | George || 3401  || Finance
|}
{{col-end}}

The antijoin is formally defined as follows:

:  {{math|1 =''R'' ▷ ''S'' = { ''t'' : ''t'' ∈ ''R'' &and; ¬&exist;''s'' ∈ ''S''(''Fun'' (''t'' &cup; ''s'')) }}}

or

:  {{math|1 = ''R'' ▷ ''S'' = { ''t'' : ''t'' ∈ ''R'', there is no tuple ''s'' of ''S'' that satisfies ''Fun'' (''t'' &cup; ''s'') }}}

where {{math|''Fun'' (''t'' &cup; ''s'')}} is as in the definition of natural join.

The antijoin can also be defined as the [[Complement (set theory)|complement]] of the semijoin, as follows:
{{NumBlk|:| {{math|1=''R'' ▷ ''S'' = ''R''&nbsp;&minus;&nbsp;''R'' ⋉ ''S''}} |{{EquationRef|5}}}}
Given this, the antijoin is sometimes called the anti-semijoin, and the antijoin operator is sometimes written as semijoin symbol with a bar above it, instead of ▷.

In the case where the relations have the same attributes (union-compatible), antijoin is the same as minus.