Editing Algebra of sets (section)

== Some additional laws for unions and intersections ==

The following proposition states six more important laws of set algebra, involving unions and intersections.

'''PROPOSITION 3''': For any [[subset]]s {{tmath|1= A }} and {{tmath|1= B }} of a universe set {{tmath|1= \boldsymbol{U} }}, the following identities hold:
: [[idempotent]] laws:
:* {{tmath|1= A \cup A = A }}
:* {{tmath|1= A \cap A = A }}
: domination laws:
:* {{tmath|1= A \cup \boldsymbol{U} = \boldsymbol{U} }}
:* {{tmath|1= A \cap \varnothing = \varnothing }}
: [[absorption law]]s:
:* {{tmath|1= A \cup (A \cap B) = A }}
:* {{tmath|1= A \cap (A \cup B) = A }}

As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws stated above<!--- in proposition 1 and proposition 2--->.  As an illustration, a proof is given below for the idempotent law for union.

''Proof:''
{|
|-
 | {{tmath|1= A \cup A }}
 | {{tmath|1= =(A \cup A) \cap \boldsymbol{U} }}
 | by the identity law of intersection
|-
 |
 | {{tmath|1= =(A \cup A) \cap (A \cup A^\complement) }}
 | by the complement law for union
|-
 |
 | {{tmath|1= =A \cup (A \cap A^\complement) }}
 | by the distributive law of union over intersection
|-
 |
 | {{tmath|1= =A \cup \varnothing }}
 | by the complement law for intersection
|-
 |
 | {{tmath|1= =A }}
 | by the identity law for union
|-
|}

The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law for union, namely the idempotent law for intersection.

''Proof:''
{|
|-
 | {{tmath|1= A \cap A }}
 | {{tmath|1= =(A \cap A) \cup \varnothing }}
 | by the identity law for union
|-
 |
 | {{tmath|1= =(A \cap A) \cup (A \cap A^\complement) }}
 | by the complement law for intersection
|-
 |
 | {{tmath|1= =A \cap (A \cup A^\complement) }}
 | by the distributive law of intersection over union
|-
 |
 | {{tmath|1= =A \cap \boldsymbol{U} }}
 | by the complement law for union
|-
 |
 | {{tmath|1= =A }}
 | by the identity law for intersection
|-
|}

Intersection can be expressed in terms of set difference:
: {{tmath|1= A \cap B = A \setminus (A \setminus B) }}