# I Quantifiers with integers and rational numbers

1. Jan 13, 2019

### ver_mathstats

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. Jan 14, 2019

### SSequence

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)]

3. Jan 14, 2019

### Staff: Mentor

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.