contestada

Translate each of the following English statements into logical expressions. The domain of discourse 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 besides 0 has a multiplicative inverse.
(h) Every number besides 0 has a unique multiplicative inverse.

Respuesta :

batolisis

Answer:

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))

Step-by-step explanation:

since the domain of discourse is a set of all real numbers the logical expressions of the English statements are expressed with respect to real number:

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))
  1. З x,y : ( x/y < 1,  y/x < 1 )
  2. ∀ Y : ( Y > 0 = 1/Y > 0 )
  3. з x,y : ( x+y = xy )
  4. ∀ x,y : [ ( x>0 ) ∧( y > 0 ) = (( x/y > 0 ) ∧ ( y/x > 0 ))
  5. ∀ y : [ ( y > 0 ) ∧ ( y < 1) = ( 1 / y > 1 )
  6. n ( ( з x ∀ y ( x <y ) )
  7. ∀x (( x ≠ 0 ) = ( зy ( xy = 1 ) ))
  8. ∀x ( (x≠0) = ( з! y (xy = 1))

In translating English written expressions to logical expressions, we have to know the domain of the discourse. The domain of discourse can be said to be a set of all real numbers that the logical expressions of the English statements are expressed with respect to their real numbers. And as such, we arrive at each of this.

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))

read more about logic here https://brainly.com/question/7581378

Otras preguntas