Boolean operators in natural language

Purpose of boolean operators

To say things such as pair ("peter","macbook") is either in relation ownsa or wantsa, requires us to use boolean operators $\cup$, $\cap$, and $-$ .

Meaning

Let us explain the meaning of relational operators $\cup$, $\cap$, and $-$ by means of examples.

Assume we have a relation, ownsa[Person*LaptopType], which contains the persons who own a particular type of laptop. A fact "peter" ownsa "macbook" means that Peter owns a MacBook.

Also assume another relation wantsa[Person*LaptopType], which contains the persons who want a particular type of laptop. A fact "peter" wantsa "macbook" means that Peter wants a MacBook.

Union

The sentence: "Peter owns a MacBook or Peter wants a MacBook." is represented as "peter" (ownsa $\cup$ wantsa) "macbook".

Intersection

The sentence: "Peter owns a MacBook and Peter wants a MacBook." is represented as "peter" (label $\cap$ colour) "macbook".

Difference

The sentence: "Peter owns a MacBook and Peter does not want a MacBook." is represented as "peter" (label $-$ colour) "macbook".

Natural language templates

There is a pattern to this. A computer can generate a literal translation from the formula to natural language. However, that translation looks clumsy, verbose and elaborate. It is up to you to turn that in normal language. For examples click here. The systematic translation is given in the following table:

Formally	Natural language template
$a\ (r\cup s)\ b$	a r b or a s b.
$a\ (r\cap s)\ b$	a r b and a s b.
$a\ (r-s)\ b$	a r b and nota s b.

Other explanation

Would you like a different explanation of the relational operators? This page explains the boolean operators in terms of set theory. An explanation in logic is given here. Click here for some algebraic rules about boolean operators. If you want to see it explained visually in Venn-diagrams, click here.