Question

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

Answer 1

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 $a \wedge b=a$