Question

Mathematical statements into logical statements with nested quantifiers. About Translate each of the following English statements into logical expressions. The domain is the set of all real numbers.
(a) There are two numbers whose ratio is less than 1.
(b) The reciprocal of every positive number is also positive.
(c) There are two numbers whose sum is equal to their product.
(d) The ratio of every two positive numbers is also positive.
(e) The reciprocal of every positive number less than one is greater than one.
(f) There is no smallest number.

Answer 1

Answer:

Answer is given below under explanation

Step-by-step explanation:

We are told that the domain is a set of real numbers. Let's depict by a & b.

We will use the following symbols;

∀ means for all

Ǝ there exists

∧ means logical conjuction

Thus;

A) Ǝ a,b : ( a/b < 1, b/a < 1 )

B) ∀ a : ( a > 0 = 1/a > 0 )

C) Ǝ a,b : (a + b = ab)

D) ∀ a,b : [ ( a > 0 ) ∧ ( b > 0 ) = (( a/b > 0 ) ∧ ( b/a > 0 ))]

E) ∀ b : [( b > 0 ) ∧ ( b < 1) = ( 1/b > 1 )]

F) n ( ( Ǝ x ∀ b ( a < b ) )