Prove that the set of all positive rational numbers has no smallest element.

Q: Q/// rational numbers is connected or disconnected in R

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Q: Prove that the sum of two rational numbers is rational

A: A rational number is a type of real number that can be expressed as the quotient or fraction of two…

Q: Prove that the set Q of rational numbers is dense in the set R of real numbers

A: Prove that the set Q of rational numbers is dense in the set R of real numbers

Q: Prove the following statement directly from the definition of rational number. The difference of any…

A: We will prove r -s is rational number by direct proof as shown below

Q: Let a,b and x be real numbers with a0. Show that there exists a rational number t such that a <

A: We know that the real numbers are densely ordered. This means that between any two distinct real…

Q: Prove that there does does not exist a unique positive real number, x, with the property that x…

A:

Q: Is this statement true or false? Prove each true statement directly from the definitions and give a…

A: Given: Given statement is The quotient of any two rational number is rational number.

Q: Let x and y be two real numbers such that x + y is rational. Prove by contrapositive that if x is…

A: Introduction: Rational and irrational numbers are two very important divisions of real number…

Q: Give an example of a countably infinite subset of the real numbers that has neither an upper bound,…

A:

Q: Prove that if x and y are rational numbers such that x &gt; y, then there exists a rational number z…

A: We have to prove that If x and y are rational numbers such that x &gt; y Then there exists a…

Q: Show that the natural numbers and the rational numbers have the same cardinality.

A: Let integers Z are countable, we might wonder whether other infinite sets are also countable. We…

Q: prove that for all positive integers therre exists two rational numbers such that there product is…

A: Let us consider two positive integers say 1 and 2. Now we find two rational numbers between them :…

Q: Prove that every algebraic integer divides a rational integer.

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Q: Is the set of rational numbers open or closed?

A: Given: Set of rational numbers. To check: Whether the set of rational numbers open or closed.

Q: All integers are rational

A: We know that "A number in the form of pq where (q≠0) is called as a rational number.

Q: Does there exist a perfect set consisting of only rational numbers?

A: Given: Does there exist a perfect set consisting of only rational numbers?

Q: 3. Prove or disprove: there exist rational numbers a and b such that a/b is irrational.

A: Suppose there exist rational numbers a and b such that a/b is irrational.

Q: If a , b are two rational numbers such that Va + vb is a rational number, then prove that Vab is…

A: We have given that , a  and  b are two rational numbers  such that  a + b  is a rational number. We…

Q: A real number of the form p/2", where p is an integer and n is a nonnegative integer, is known as a…

A: We have to prove that there is a dyadic rational number between any two distinct real numbers. By…

Q: Prove that Between any two rational numbers there exist an irrational number .

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.