Let ( L, ≤ ) be a lattice, x, y, z ∈ L
I am unable to understand why ( x ∧ z ) ∧ ( y ∧ z ) = z ∧ ( y ∧ x ) is mentioned as fact in Y.N. Singh's "Mathematical Foundation of Computer Science" - Pg 157
Could I prove this somehow, or is it an axiom?
Let ( L, ≤ ) be a lattice, x, y, z ∈ L
I am unable to understand why ( x ∧ z ) ∧ ( y ∧ z ) = z ∧ ( y ∧ x ) is mentioned as fact in Y.N. Singh's "Mathematical Foundation of Computer Science" - Pg 157
Could I prove this somehow, or is it an axiom?
There are at least two ways to do this:
As Robin Chapman suggests, show that each side of the equation is just the infimum of the set $\{x,y,z\}$
Prove that $\wedge$ is an associative and commutative operation, and then use this to reduce the left side to $x \wedge y \wedge (z \wedge z)$ which is $x \wedge y \wedge z$ because $z \wedge z = z$.