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4.1: Quantifiers

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    23899
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    Earlier, we observed that cannot fully express ideas involving quantity, such as “some” or “all.” In this chapter, we will fill this gap by introducing quantifier symbols. Together with predicates and sets, which have already been discussed, this completes the language of . We will then use this language to translate assertions from English into mathematical notation.

    In our first example we will use this symbolization key:

    \(\mathcal{U}\): The set of all people.
    \(L\): The set of all people in to Lethbridge.
    \(A\): The set of all angry people.
    \(H\): The set of all happy people.
    \(x\) \(R\) \(y\): \(x\) is richer than \(y\)
    \(d\): Donald
    \(g\): Gregor
    \(m\): Marybeth

    Now consider these assertions:

    1. Everyone is happy.
    2. Everyone in Lethbridge is happy.
    3. Everyone in Lethbridge is richer than Donald.
    4. Someone in Lethbridge is angry.

    It might be tempting to translate Assertion 1. as \((d \in H) \& (g \in H) \& (m \in H)\). Yet this would only say that Donald, Gregor, and Marybeth are happy. We want to say that everyone is happy, even if we have not listed them in our symbolization key. In order to do this, we introduce the “\(\forall\)” symbol. This is called the universal quantifier. \[\forall x \text { means "for all } x \text { " }\]

    A quantifier must always be followed by a variable, and then a formula that the quantifier applies to. We can translate Assertion 1. as \(\forall x, (x \in H)\). Paraphrased in English, this means “For all \(x\), \(x\) is happy.”

    In quantified assertions such as this one, the variable \(x\) is serving as a kind of placeholder. The expression \(\forall x\) means that you can pick anyone and put them in as \(x\). There is no special reason to use \(x\) rather than some other variable. The assertion “\(\forall x, (x \in H)\)” means exactly the same thing as “\(\forall y , (y \in H)\),” “\(\forall z, (z \in H)\),” or “\(\forall x_5, (x_5 \in H)\).”

    To translate Assertion 2., we use a different version of the universal quantifier: \[\text { If } X \text { is any set, then } \forall x \in X \text { means "for all } x \text { in } X \text { " }\]
    Now we can translate Assertion 2. as \(\forall \ell \in L, (\ell \in H)\). (It would also be logically correct to write \(\forall x \in L, (x \in H)\), but \(\ell\) is a better name an element of the set \(L\).) Paraphrased in English, our symbolic assertion means “For all \(\ell\) in Lethbridge, \(\ell\) is happy.”

    Assertion 3. can be paraphrased as, “For all \(\ell\) in Lethbridge, \(\ell\) is richer than Donald.” This translates as \(\forall \ell \in L, (\ell \mathrel{R} d)\).

    To translate Assertion 4., we introduce another new symbol: the existential quantifier, \(\exists\). \[\exists x \text { means "there exists some } x \text {, such that" }\]

    If \(X\) is any set, then \(exists x \in X\) means
    "there exists some \(x\) in \(X\), such that"

    We write \(\exists \ell \in L, (\ell \in A)\). This means that there exists some \(\ell\) in Lethbridge who is angry. More precisely, it means that there is at least one angry person in Lethbridge. Once again, the variable is a kind of placeholder; it would have been logically correct (but poor form) to translate Assertion 4. as \(\exists z \in L, (z \in A)\).

    Example \(4.1.1\).

    Consider this symbolization key.

    \(S\): The set of all students.
    \(B\): The set of all books.
    \(N\): The set of all novels.
    \(x\) \(L\) \(y\): \(x\) likes to read \(y\).

    Then:

    1. \(\forall n \in N, (n \in B)\) means “every novel is a book,” and
    2. \(\forall s \in S, \bigl( \exists b \in B, (s \mathrel{L} b) \bigr)\) means “for every student, there is some book that the student likes to read.”

    Notice that all of the quantifiers in this example are of the form \(\forall x \in X\) or \(\exists x \in X\), not \(\forall x\) or \(\exists x\). That is, all of the variables range over specific sets, rather than being free to range over the entire universe of discourse. Because of this, it is acceptable to omit specifying a universe of discourse. Of course, the universe of discourse (whatever it is) must include at least all students, all books, and all novels.

    Exercise \(4.1.2\).

    Suppose \(A\) and \(B\) are sets.
    Give your answers in the notation of First-Order Logic (not English).

    1. What does it mean to say that \(A\) is a subset of \(B\)?
    2. What does it mean to say that \(A\) is not a subset of \(B\)?

    This page titled 4.1: Quantifiers is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Dave Witte Morris & Joy Morris.

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