Translate the following English sentence into a symbolic sentence with quantifiers:
Between any integer and any larger integer, there is a real number. (universe is real numbers)
Check the following:
∀a∈ℤ∀(b∈ℤ)>a∃r∈ℝ(a<r<b)
To translate the given sentence into a symbolic sentence with quantifiers, we can break it down into smaller parts and assign variables to represent the different elements.
Let's define the universe as the set of real numbers, denoted as R.
Now, let's assign the following variables:
- x: represents an integer
- y: represents a larger integer
- z: represents a real number
The statement "Between any integer and any larger integer, there is a real number" can be translated into symbolic logic as follows:
∀x∀y∃z (x < z < y)
Explanation of the translation:
- ∀x: This quantifier (∀) represents the universal quantification of x, meaning "for all x" or "for any integer."
- ∀y: This quantifier (∀) represents the universal quantification of y, meaning "for all y" or "for any larger integer."
- ∃z: This quantifier (∃) represents the existential quantification of z, meaning "there exists a z" or "there is a real number."
- (x < z < y): This is the predicate that represents the condition "x is less than z, and z is less than y." It ensures that z lies between x and y, indicating the existence of a real number between any integer (x) and any larger integer (y).
So, the translated symbolic sentence is:
"For any integer x and any larger integer y, there exists a real number z such that x < z < y."