Confusion about notation regarding compliments/universal sets

[NecroWinter]

Main Question or Discussion Point

Im not sure how to interpret the notation, specifically the one on the left, the one on the right seems like you just include everything in the universal set?

what does it mean when the line goes over everything? what does it even mean when the line is over the and/or symbol

 

[Stephen Tashi]

Using the line is similar to using parentheses with statements.

For example, the meaning of " not in ( A or B)" is different than the meaning of "(not in A) or (not in B)"

Suppose a vase V is (not alabaster) and is blue.

Then it is is true that "(V is not alabaster) or (V is not blue)"
because V is not alabaster.

It is false that "V is not (alabaster or blue)"
because V is blue, hence it IS (alabaster or blue).

 

[Akshay_Anti]

let us first understand the basic concepts involved-

A,B are the name of the sets involved

A U B refers to a collection that includes all elements that occur in A and (or) B i.e. those that occur in A only + those that occur in B only + those that occur in both of them

A bar over a set means that we've to take negation of that set.

$\cap$ symbol means intersection and considers only those elements that are common to both the sets

now, a bar over whole of ( A U B ) means that we have to take those elements in father or universal set defined for the two sets which don't occur in the set ( A U B )

and the part b refers to the elements which are common between ( the elements contained in father set but not in set A ) and ( the elements contained in father set but not in set B )