4) Show that the set of all numbers that are not solutions of polynomial equations with...
3) Show that the set of al numbers that are solutions of polynomial equations with integer coefficients is a countable set.
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)...
2. (Countable and uncountable sets and diagonalization) (a) A polynomial in variable x is an expression of the form c0 + c1x + c2x2 +c3x3 +· · ·+cdxd , where d is a non-negative integer and c0, · · · , cd are constants, called coefficients. Let P be the set of polynomials with integer coefficients. Show that P is countable.
please help me with this assignment and show all the work. thank
you!
Determine if each set countable or uncountable. Show a proof or argument to justify your decision for each set. a) the ages of students in this class b) the integers that are multiples of 10 c) Real numbers between and including 4 and 6 [4,6]
2) Show that the set that contains all the subsets of the natural numbers N (i.e. the power set of N usually denoted by 2) is uncountable.
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.
Let X be a set and let T be the family of subsets U of X such
that X\U (the complement of U) is at most countable, together with
the empty set. a) Prove that T is a topology for X. b) Describe the
convergent sequences in X with respect to this topology. Prove that
if X is uncountable, then there is a subset S of X whose closure
contains points that are not limits of the sequences in S....
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
Let X be the set of all 2 x 2 matrices -- SRC HES where a and b are integer numbers, while c and d are rational numbers. Prove that the set X is countable.
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z...