Question

Exercise 1.10.4: Mathematical statements into logical statements with nested quantifiers. info 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. (g) Every number other than 0 has a multiplicative inverse. (h) Every number other than 0 has a unique multiplicative inverse.

Answer 1

З x,y : ( x/y < 1, y/x < 1 )

Explanation:

This is a bit somehow, I really do hope it doesn't confuse you. Here's your solution.

The domain in this question is one with a set of all real numbers. These real numbers needs the logical expressions of the English statements to be expressed with respect to real number. And so we're having

A) З x,y : ( x/y < 1, y/x < 1 )

B) ∀ Y : ( Y > 0 = 1/Y > 0 )

C) з x,y : ( x+y = xy )

D) ∀ x,y : [ ( x>0 ) ∧( y > 0 ) = (( x/y > 0 ) ∧ ( y/x > 0 ))

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

F) n ( ( з x ∀ y ( x <y ) )

G) ∀x (( x ≠ 0 ) = ( зy ( xy = 1 ) ))

H) ∀x ( (x≠0) = ( з! y (xy = 1))