prove that the set of all algebraic number is countable
6. (5 pts.) A real number r is called an algebraic number if r is a zero of a polynomial Plx)=a,x" +a,-|x"-, + +a,x + ao with integer coefficients. Prove that the set A of all algebraic numbers is countable.
13. An algebraic number is a real number which is the root of a polynomial co + ciz c2n in which all of the coefficients c i 1,2,.,n) are integers. The order of an algebraic number is the smallest natural number n for which z is a root of an n-th degree polynomial with integer coefficients. A real number is transcendental if it is not algebraic. a) Show that the set of algebraic numbers of order n is countable (b)...
Problem 6 Suppose A and B are countable sets. Prove A × B is a countable set.
3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from class. (You need to prove that your function is a bijection.) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Problem 1. Let A C R be a countable set. Prove that R\ A is uncountable.
3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a set G (a countable intersection of open sets), and a set F (a countable union of closed sets) such that F CE C G and m* (F) the Lebesgue measure of a set Hint: The Lebesgue measure can be calculated in terms of open and closed sets m* (E) m* (G), where m* denotes 3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a...
12) Let S1 be a countable set, S2 a set that is not countable, and S1 ⊂ S2. a) Show that S2 must then contain an infinite number of elements that are not in S1. b) Show that in fact S2 − S1cannot be countable.
Explain or prove your answer. Is the following set finite, countable or uncountable? {(x, y) E NXR : xy = 1}
1. Prove that any infinite set contains a countable subset (see Problem 20, page 43)
a set (any set of objects) is said to be countable if it is either finite or there is an enumeration (list) of the set. show that the following properties hold for arbitrary countable sets: a) All subsets of countable sets are countable b) any union of a pair of countable sets is countable c) all finite sets are countable