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Home / Discrete mathematics / Absorption Property, Idempotent Property| Discrete mathematics

Absorption Property, Idempotent Property| Discrete mathematics

Absorption Property: For any a, b in A, a˅(a˄b) =a and a˄(a˅b)=a

Proof: Since a˅(a˄b) is the join of a and (a˄b) then ,

a˅(a˄b) ≥a ————————-(1)

Since a≥a and a≥(a˄b) which implies that

 

(a˅a) ≥ a˅(a˄b)

⇒ a≥ a˅(a˄b) ———————————(2)

From (1) and (2) we get,

a˅(a˄b)=a

Now by the principle of duality we get,

a˄(a˅b)=a                                                         (proved)

Idempotent Property: For every a in A, (a˅a) =a and (a˄a)=a

Proof: We have, a≤(a˅a) ————————————–(1)

i.e., join of any two elements in a lattice is greater or equal to each of the elements.

Since, a≤a

Therefore, (a˅a) ≤a —————————————————–(2)

[if a≤b and a≤a then (a˅c) ≤b is because (a˅c) is the least upper bound of a and c.]

From (1) and (2) we get

(a˅a) =a

Now by the principle of duality, we get,

(a˄a)=a                                                                                         (proved)

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