Algebra of Sets
Suppose that $A,B,C,U$ are sets with $A \subseteq U, B \subseteq U, C \subseteq U$.
Commutative Laws
These laws say that is doesn’t matter in which order the operation takes place.
- $A\cup B = B \cup A$
- $A\cap B = B \cap A$
Associative Laws
These laws say that is doesn’t matter where you put brackets around an operation
- $A\cup(B\cup C)=(A\cup B) \cup C$
- $A \cap(B \cap C) = ( A \cap B) \cap C$
Distributive Laws
This is how the properties of union and intersect propagate between one another. This applies to expanding brackets with mixed signs.
- $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$
- $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$
Identity Laws
The following sets only apply if there is a universe set or context which the sets are within.
If you join the empty set with anything you get what you joined it with. If you take the intersect of anything and the universe $U$ then you get the thing you intersected it with.
- $A\cup \emptyset = A$
- $A\cap U = A$
Complement Laws
If you take a set and it’s complement (¬) then you get the universe set $U$. The intersect of a set and its complement is the empty set.
- $A\cup\sim A = U$
- $A\cap\sim A=\emptyset$
Double Complement Law
- $\sim(\sim A ) = A$
Idempotent Laws
- $A \cup A = A$
- $A \cap A = A$
Universal Bound Laws
- $A\cup U = U$
- $A\cap \emptyset = \emptyset$
De Morgan’s Law
The complement of a union is the intersection of complements.
- $\sim(A\cup B) = \sim A\cap\sim B$
- $\sim(A\cap B) = \sim A\cup\sim B$
Absorption Laws
This is a result of the expansion of brackets.
- $A\cup(A\cap B) = A$
- $A\cap(A\cup B) = A$
Complement of $U$ and $\emptyset$
- $\sim U = \emptyset$
- $\sim \emptyset = U$
Set Difference Law
- $A-B=A\cap\sim B$
Disclaimer and Proofs
As none of the above are axioms then you have to reference them if you use them in a proof.