Question

Rewrite the following statement formally. Use variables and include both quantifiers for all & existential in your answer: Every rational number can be written as s a ration of some two integers.

a) For all x in Q, there exist a, b in Z such that x = a/b
b) There exists x in Q such that for all a, b in Z, x = a/b
c) For all a, b in Z, there exists x in Q such that x = a/b
d) There exists a, b in Z such that for all x in Q, x = a/b

Answer 1

The correct formal statement reflecting that every rational number can be written as a ratio of two integers is a) 'For all x in Q, there exist a, b in Z such that x = a/b'. This uses the correct quantifiers and is the only option that properly conveys the definition of a rational number.

Step-by-step explanation:

The question at hand involves understanding how rational numbers can be formally expressed using variables and quantifiers. By definition, a rational number is any number that can be written as the ratio of two integers, a and b, where b is not zero. When rewriting the given statement formally, it is crucial to use the correct logical quantifiers to accurately express this definition. Now, let's look at our options.

Option a) 'For all x in Q, there exist a, b in Z such that x = a/b' correctly uses the universal quantifier (for all) to establish that this is true for every rational number x in the set of rational numbers, Q. It then uses the existential quantifier (there exist) to indicate that for each x there are some integers a and b that form the ratio a/b, which equals x. This accurately reflects the definition of a rational number.

Option b) 'There exists x in Q such that for all a, b in Z, x = a/b' incorrectly suggests that there is only one rational number x that can be written as a ratio of any integers a and b, which is not true.

Option c) 'For all a, b in Z, there exists x in Q such that x = a/b' inaccurately implies that for every pair of integers, there is a different rational number, which is also not correct.

Option d) 'There exists a, b in Z such that for all x in Q, x = a/b' implies that there is a single ratio of two integers that equals every rational number, which is clearly incorrect.

Therefore, the correct option that reflects the original statement about rational numbers is option a): 'For all x in Q, there exist a, b in Z such that x = a/b'. This formulation correctly uses both the universal and existential quantifiers to indicate that every rational number is a ratio of two integers.