Le të supozojmë se x është cilido një element i bashkësisë (A
∪
{\displaystyle \scriptstyle {\cup }}
x
∈
{\displaystyle \scriptstyle \in }
∪
{\displaystyle \scriptstyle {\cup }}
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∉
{\displaystyle \scriptstyle \not \in }
∪
{\displaystyle \scriptstyle {\cup }}
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∉
{\displaystyle \scriptstyle \not \in }
∧
{\displaystyle \scriptstyle \land }
∉
{\displaystyle \scriptstyle \not \in }
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∈
{\displaystyle \scriptstyle \in }
∧
{\displaystyle \scriptstyle \land }
∈
{\displaystyle \scriptstyle \in }
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∈
{\displaystyle \scriptstyle \in }
∩
{\displaystyle \scriptstyle {\cap }}
Meqë, implikacioni x
∈
{\displaystyle \scriptstyle \in }
∪
{\displaystyle \scriptstyle {\cup }}
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∈
{\displaystyle \scriptstyle \in }
∩
{\displaystyle \scriptstyle {\cap }}
(A
∪
{\displaystyle \scriptstyle {\cup }}
(
∀
{\displaystyle \scriptstyle {\forall }}
∈
{\displaystyle \scriptstyle \in }
∪
{\displaystyle \scriptstyle {\cup }}
∈
{\displaystyle \scriptstyle \in }
∪
{\displaystyle \scriptstyle {\cup }}
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∈
{\displaystyle \scriptstyle \in }
∩
{\displaystyle \scriptstyle {\cap }}
konkludojmë se (A
∪
{\displaystyle \scriptstyle {\cup }}
∩
{\displaystyle \scriptstyle {\cap }}
(2) Vërtetimi i inkluzionit A'
∩
{\displaystyle \scriptstyle {\cap }}
∪
{\displaystyle \scriptstyle {\cup }}
Le të supozojmë tani se y A'
∩
{\displaystyle \scriptstyle {\cap }}
y
∈
{\displaystyle \scriptstyle \in }
∩
{\displaystyle \scriptstyle {\cap }}
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∈
{\displaystyle \scriptstyle \in }
∧
{\displaystyle \scriptstyle \land }
∈
{\displaystyle \scriptstyle \in }
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∉
{\displaystyle \scriptstyle \not \in }
∧
{\displaystyle \scriptstyle \land }
∉
{\displaystyle \scriptstyle \not \in }
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∉
{\displaystyle \scriptstyle \not \in }
∪
{\displaystyle \scriptstyle {\cup }}
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∈
{\displaystyle \scriptstyle \in }
∪
{\displaystyle \scriptstyle {\cup }}
Meqë edhe këtu implikacioni y
∈
{\displaystyle \scriptstyle \in }
∩
{\displaystyle \scriptstyle {\cap }}
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∈
{\displaystyle \scriptstyle \in }
∪
{\displaystyle \scriptstyle {\cup }}
A'
∩
{\displaystyle \scriptstyle {\cap }}
(
∀
{\displaystyle \scriptstyle {\forall }}
∈
{\displaystyle \scriptstyle \in }
∩
{\displaystyle \scriptstyle {\cap }}
∈
{\displaystyle \scriptstyle \in }
∩
{\displaystyle \scriptstyle {\cap }}
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
∈
{\displaystyle \scriptstyle \in }
∪
{\displaystyle \scriptstyle {\cup }}
konkludojmë se A'
∩
{\displaystyle \scriptstyle {\cap }}
∪
{\displaystyle \scriptstyle {\cup }}
(3) Nga inkluzionet të vërtetuara nën (1) dhe (2) dhe në bazë të përkufizimit 2.1 .3. :
(A
∪
{\displaystyle \scriptstyle {\cup }}
∩
{\displaystyle \scriptstyle {\cap }}
}
{\displaystyle \displaystyle {\}}}
⇔
{\displaystyle \scriptstyle \Leftrightarrow }
∪
{\displaystyle \scriptstyle {\cup }}
=
{\displaystyle \scriptstyle {=}}
∩
{\displaystyle \scriptstyle {\cap }}
A'
∩
{\displaystyle \scriptstyle {\cap }}
∪
{\displaystyle \scriptstyle {\cup }}
Fig. 1.6.
konkludojmë se është i saktë relacioni që shpreh ligjin e parë të De Morganit . Në mënyrë analoge bëhet vërtetimi i ligjit të dytë [ 1] Le të jenë A, B A\B B .A diferenca simetrike e tyre dhe shënohet A
∇
{\displaystyle \scriptstyle {\nabla }}
A
∇
{\displaystyle \scriptstyle {\nabla }}
=
{\displaystyle \scriptstyle {=}}
∪
{\displaystyle \scriptstyle {\cup }}
P.sh. : {a, b, c}
∇
{\displaystyle \scriptstyle {\nabla }}
=
{\displaystyle \scriptstyle {=}}
∪
{\displaystyle \scriptstyle {\cup }}
=
{\displaystyle \scriptstyle {=}}
∪
{\displaystyle \scriptstyle {\cup }}
=
{\displaystyle \scriptstyle {=}}
Pra, diferenca simetrike e bashkësive A, B
↑ Vërtetimi i ligjeve të De Morganit shkurtohet nëse në vend të
⇒
{\displaystyle \scriptstyle {\Rightarrow }}
⇔
{\displaystyle \scriptstyle \Leftrightarrow }
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