CMPS 260
Some Laws of Sets and Logic

Axioms:
Union: x ∈ S ∪ T ≡ x ∈ S ∨ x ∈ T
Intersection: x ∈ S ∩ T ≡ x ∈ S ∧ x ∈ T
Difference: x ∈ S − T ≡ x ∈ S ∧ x ∈ T
Complement: x ∈ S ≡ x ∉ S
Non-member: x ∉ S ≡ ¬(x ∈ S)
Equality: S = T ≡ x ∈ S = x ∈ T (for all x)
Subset: S ⊆ T ≡ x ∈ S ⟹ x ∈ T (for all x)

Theorems:
DeMorgan (1): S ∩ T = S ∪ T
DeMorgan (2): S ∪ T = S ∩ T

Some Laws of Propositional Logic

Double Negation: ¬¬p≡ p
Implication: p ⟹ q≡ ¬p ∨ q
Contrapositive: p ⟹ q≡ ¬q ⟹ ¬p
DeMorgan (1): ¬(p ∧ q)≡ ¬p ∨ ¬q
DeMorgan (2): ¬(p ∨ q)≡ ¬p ∧ ¬q

Sample proof of DeMorgan (1) (with respect to sets):

By the Equality Axiom, to show that two sets are equal it suffices to show that, for arbitrary x, x is a member of one set iff it is a member of the other. That is what we do below: 

     x ∈ S ∩ T 

=       < Complement >

     x ∉ S ∩ T

=       < Non-member >

     ¬(x ∈ S ∩ T)

=       < Intersection >

     ¬(x ∈ S ∧ x ∈ T)

=       < DeMorgan (1) (propositional) >

     ¬(x ∈ S) ∨ ¬(x ∈ T)

=       < Non-member >

     x ∉ S  ∨  x ∉ T

=       < Complement >

     x ∈ S ∨ x ∈ T 

=       < Union >

     x ∈ S ∪ T