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Archangel Macsika

Let L be a lattice. Then prove that a Ù b=a if and only if a v b=b.

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Let (L,\vee, \wedge) be a lattice. We want to prove that a \wedge b=a if and only if a\vee b=b

Suppose a \wedge b =a, since a \wedge b \leq b. Thus, a \leq b

if a \leq b , since b \leq b , thus b is a upper bound of a and b , by definition of least upper bound we have a \vee b \leq b . since a \vee b is an upper bound of a and b ,b \leq a \vee b , so a \vee b=b


Suppose a \vee b =b, since a \vee b \leq b. Thus, b \leq a

if a \leq a , since b \leq a , thus a is a upper bound of a and b , by definition of least upper bound we have a \wedge b \leq a . since a \vee b is an upper bound of a and b ,a \leq a \wedge b , so a \wedge b=a


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