0.3 Discrete structures logic  (Page 13/23)

Similarly the existential quantifier turns, for example, the statement x>1 to "for some object x in the universe, x>1", which is expressed as "∃x x>1." Again, it is true or false in the universe of discourse, and hence it is a proposition once the universe is specified.

Universe of discourse

The universe of discourse, also called universe, is the set of objects of interest. The propositions in the predicate logic are statements on objects of a universe. The universe is thus the domain of the (individual) variables. It can be the set of real numbers, the set of integers, the set of all cars on a parking lot, the set of all students in a classroom etc. The universe is often left implicit in practice. But it should be obvious from the context.

The universal quantifier

The expression: ∀x P(x), denotes the universal quantification of the atomic formula P(x). Translated into the English language, the expression is understood as: "For all x, P(x) holds", "for each x, P(x) holds" or "for every x, P(x) holds". ∀is called the universal quantifier, and ∀x means all the objects x in the universe. If this is followed by P(x) then the meaning is that P(x) is true for every object x in the universe. For example, "All cars have wheels" could be transformed into the propositional form, ∀x P(x), where:

• P(x) is the predicate denoting: x has wheels, and

• the universe of discourse is only populated by cars.

Universal quantifier and connective and

If all the elements in the universe of discourse can be listed then the universal quantification ∀x P(x) is equivalent to the conjunction: P(x1)) ⋀P(x2) ⋀P(x3) ⋀... ⋀P(xn) . For example, in the above example of ∀x P(x), if we knew that there were only 4 cars in our universe of discourse (c1, c2, c3 and c4) then we could also translate the statement as: P(c1) ⋀P(c2) ⋀P(c3) ⋀P(c4) .

The existential quantifier

The expression: ∃xP(x), denotes the existential quantification of P(x). Translated into the English language, the expression could also be understood as: "There exists an x such that P(x)" or "There is at least one x such that P(x)" ∃ is called the existential quantifier, and ∃x means at least one object x in the universe. If this is followed by P(x) then the meaning is that P(x) is true for at least one object x of the universe. For example, "Someone loves you" could be transformed into the propositional form, ∃x P(x), where:

• P(x) is the predicate meaning: x loves you,

• The universe of discourse contains (but is not limited to) all living creatures.

Existential quantifier and connective or

If all the elements in the universe of discourse can be listed, then the existential quantification ∃xP(x) is equivalent to the disjunction: P(x1) ⋁P(x2) ⋁P(x3) ⋁... ⋁P(xn).

For example, in the above example of ∃x P(x), if we knew that there were only 5 living creatures in our universe of discourse (say: me, he, she, rex and fluff), then we could also write the statement as: P(me)⋁P(he) ⋁P(she) ⋁P(rex) ⋁P(fluff)

An appearance of a variable in a wff is said to be bound if either a specific value is assigned to it or it is quantified. If an appearance of a variable is not bound, it is called free. The extent of the application (effect) of a quantifier, called the scope of the quantifier, is indicated by square brackets [ ]. If there are no square brackets, then the scope is understood to be the smallest wff following the quantification.