Universal Quantifiers

Learn about universal quantifiers and how they are used with predicates to produce powerful mathematical statements.


We make a proposition from a predicate by assigning a particular value to its variable(s). The value we can give to a variable is from a domain. We can also make a proposition from a predicate without assigning a particular value to its variable. One way to do so is by referring to all the domain elements. Universally quantifying a predicate means that the proposition resulting from that predicate is true for every element of the domain. Therefore, we get a proposition by applying a universal quantifier to a predicate. The motivation is to make propositions without referring to a particular domain element.

Universal quantifier

We use universal quantifiers to claim that every domain member has the property mentioned by a predicate. They quantify a variable in a predicate for a particular domain. To say that every student in discrete math class has passed twelfth grade or equivalent, we make a predicate T(x)T(x).

  • T(x)T(x): xx has passed twelfth grade or equivalent.

To say that every xx in discrete math class has the property TT, we need to quantify it. We use the symbol \forall to represent universal quantification, and we read this symbol as “for all.”

  • xT(x)\forall x T(x): For all x,T(x)x, T(x).

The domain of this quantification is the discrete math class. The meaning of T(x)T(x) is as above, but the part, “x\forall x,” requires a domain to make sense. Once the domain is explicit, \forall refers to each member of it.

Assume the domain, DD, has nn elements. That is,

D={e1,e2,e3,,en}.D = \{e_1,e_2,e_3, \ldots, e_n\}.

For some arbitrary predicate PP, we can interpret xP(x)\forall x P\left(x\right) as follows:

xP(x)=P(e1)P(e2)P(e3)P(en).\forall x P(x) = P(e_1)\land P(e_2)\land P(e_3)\land \ldots \land P(e_n).

That means xP(x)\forall x P(x) is true if and only if P(x)P(x) is true for every domain element. This way, we can intuitively think about the universal quantifier as a big conjunction of instantiated predicates.

If a predicate has more than one variable, for example, P(x,y,z)P(x,y,z), we can universally quantify each of them as follows:

  • xyzP(x,y,z):\forall x \forall y \forall z P(x,y,z): For all xx and for all yy and for all zz, P(x,y,z).P(x,y,z).

Truth value

The statement xP(x)\forall x P(x) is true when every element from the considered domain has the property PP. It is false if we can show at least one element from the domain that does not have the property PP. We call such an element a counterexample.

Let’s look at a few examples to clarify further the concept of universal quantification and the truth value of quantified statements.


Let’s take a look at a few examples.

Consider the following predicate:

  • S(x)S(x): The student xx is at least 16 years old.

Assume that the domain is the department of computer science at Oxford University. We can quantify S(x)S(x) as follows:

  • xS(x)\forall x S(x): Every student (in the computer science department) is at least 16 years old.

Be cautious about the domain while interpreting this statement; it can change the meaning and the truth value of the statement.

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