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I Quantifiers with integers and rational numbers

  1. Jan 13, 2019 #1
    Give an example where a proposition with a quantifier is true if the quantifier ranges over the integers, but false if it ranges over rational numbers.

    I do not know where to go about when answering this, I know that an integer can be a rational number, for example 5 is an integer but can also be turned into 5/1 thus becoming a rational number.

    However I do not know where to go from there.
  2. jcsd
  3. Jan 14, 2019 #2
    There could be many examples. One example that I can think of is that:
    For any two numbers x,y∈A (such that x<y) there exists a number z∈A such that x<z<y

    The above statement will be true for rational numbers, but false for natural numbers and integers.

    I think we can write it (somewhat informally) as something like:
    ∀x∀y∃z[(x<y)→(z>x and z<y)]
  4. Jan 14, 2019 #3


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    exists: There exist prime numbers in ##\mathbb{Z}## but none in ##\mathbb{Q}##.
    for all: The minimum distance between two distinct integers is one, whereas this is not true for rationals.
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