Partial order relations, on boolean algebra

[buchi]

I am a bit confused about a question on proving partial order relation. here is the question and what i done so far.

"define the relation '≤' on a boolean algebra B by
for all x,yεB x≤y if and only if xVy=y, show that '≤' is a partial order relation"

first of all what exactly does boolean algebra B look like? can you give me an example of a set A that is a boolean algebra?

I have done a lot of examples to prove equivalence relation last week and the idea is straight forward and with this one I first tried to prove reflexivity antisymmetry and transitive



reflexivity:
for any element x that is in B x≤x that is xVx=x? this part doesn't make sense to me nor do I know what does xVx mean x or x as a set operation 'OR' if so could you guys lead me in a bit

I think once i understand what exactly xVy=y means and how i can manupulate it i think i will be ok to prove antisymmetry transitivity and reflexivity

thanks guys


thanks in advance

 

[Valkarie]

A boolean algebra is a set A that is closed under the operations of union (∪), intersection (∩), and complement (ˉ). As an example, consider the set A = {a,b,c,d}. Then A is a boolean algebra because it is closed under the operations of union, intersection and complement. To prove reflexivity for this relation, we need to show that for any element x in B, x ≤ x. Since xVy=y, this means that if x is equal to y, xVy must be equal to y. Hence, for any element x in B, xVx=x and so x ≤ x, which proves that the relation is reflexive. To prove antisymmetry, we need to show that for any elements x, y in B, if x ≤ y and y ≤ x, then x must be equal to y. Since xVy=y, this means that if x is equal to y, xVy must be equal to y. Hence, if x ≤ y and y ≤ x, then xVy must be equal to both x and y. This implies that x must be equal to y, which proves that the relation is antisymmetric.Finally, to prove transitivity, we need to show that for any elements x, y, z in B, if x ≤ y and y ≤ z, then x ≤ z. Since xVy=y and yVz=z, this means that if x is equal to y and y is equal to z, then xVz must be equal to z. Hence, if x ≤ y and y ≤ z, then xVz must be equal to z, which proves that the relation is transitive.