Prove the complementation law by showing A=A
(assume that A is a subset of some underlying
universal set U)
A is the complement of A. That is, A={x∈U∣x∈/A}. x∈/A means x∈A. Represented differently, A={x∣¬x∈A}={x∣¬¬x∈A}={x∣x∈A}=A.
Prove the domination laws by showing
a) A∪U=U
b) A∩∅=∅
(assume that A is a subset of some underlying
universal set U)
a) A∪U={x∣x∈A∨x∈U}={x∣x∈A∨T}={x∣T}=U.
b) A∩∅={x∣x∈A∧x∈∅}={x∣x∈A∧F}={x∣F}=∅
Prove the complement laws by showing that
a) A∪Aˉ=U
b) A∩Aˉ=∅
(assume that A is a subset of some underlying
universal set U)
a) A∪Aˉ={x∣x∈A∨x∈Aˉ}={x∣x∈A∨¬x∈A∧x∈U}={T∧x∈U}={x∈U}=U
(This one uses negation and identity laws)
b) A∩Aˉ={x∣x∈A∧x∈A}={x∣x∈A∧¬x∈A}={x∣F}=∅