please explain

Question

every total ordered set is distributive.

Answer 1

Every totally ordered set is a distributive lattice. 

A totally ordered set means, for every two elements of the set $a$ and $b$, either $a \leq b$ or $b \leq a$, and thus $a \leq b$ if and only if $a \wedge b = a$ (meet operation) and $a \vee b = b$ (join operation). 

A lattice $(L,∨,∧)$ is distributive if the following identity holds for all $x, y,$ and $z$ in $L$:

$$x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)$$.

$\text{LHS} = x \wedge (y \vee z)$.

For a totally ordered set, this will return $x$, iff order of $x$ is $\leq$ either $y$ or $z$. Otherwise it will return the larger of $y$ and $z$.

The same thing is applicable for $\text{RHS}$ also making $\text{LHS} = \text{RHS}$ for a totally ordered set.

Comments

Thank you Sir

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Suppose y ≤ x ≤ z    Here , I m getting x∧(y∨z)  = X ^ Z = X

Then how this statement is true :

              this will return x, iff order of  x is ≤ both y and z ??
 

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Thanks - corrected..