Let a and b are two distinct real numbers. Show that if a < b, there...
A. Let a and c be real numbers, with a<c. Using the axioms of the real number system, prove there exists a real number b so that a<b<c.
Let f: [a, b] → [a,b] be a continuous function, where a, b are real numbers with a < b. Show that f has a fixed point (i.e., there exists x e [a, b] such that f(x) = x).
3. Let a, b, and c be real numbers, with c +0. Show that the equation ax2 + bx + c = 0 (a) has two (different) real solutions if 62 > 4ac, (b) has one real solution if 62 = 4ac, and (c) has two complex conjugate solutions if 62 < 4ac.
[9] Given any two real numbers x and y such that x < y, show that there exists a rational number q such that x < a <y.
pleases show all steps thankyou and write clearly Let a and b be rational numbers, and an irrational number. Let S be the statement t aH b then is irrational b+i ()Write down the converse and the contrapositive of S >p (i) Converse: & (ii) Contrapositive: a 77 bAX (b) Write down a proof that S is true.
I need help with Problem 11. Thanks! 3 of 3 11. Result: There exists two distinct (different) irrational numbers a and b such that a' is rational 12. Result: 24(5 -1) for every positive integer n.
.3. Let A and B be distinct points. Prove that for each real number r E (-00, oo) there is exactly one point on the extended line AB such that AX/XB- r. Which point on AB does not correspond to any real number r? 4. Draw an example of a triangle in the extended Euclidean plane that has one ideal vertex. Is there a triangle in the extended plane that has two ideal vertices? Could there be a triangle with...
Let AC (0,1) be the set of real numbers with a decimal expansion containing only Os, 2s, and 5s. For example, 2/9 = 0.222... € A and 0.2500525... E A, but 1/8 = 0.125 € A. Prove that A is uncountable. Let A = {a,b,c,r,s.t} be a set with 6 distinct elements. Either construct a binary operation f: AxA+A with the property that for every 2 EA, fía, 2) = 2, f(1, ) = , and f(0, 2) = 2,...
Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are similar over C, then they are also similar over IK (a) Prove that for all X, y є Rnxn, the function f(t) det (X + ty) is a polynomial in t. (b) Prove that if X and Y are real n × n matrices such that X + ừ is an invertible complex matrix, then there exists a...
in R × R, for any two distinct points A and B, there exists a unique line containing them Show this statement is not true in Z, x Z, by finding the equations of two distinct lines that both pass through the points (1,2) and (3,4) in R × R, for any two distinct points A and B, there exists a unique line containing them Show this statement is not true in Z, x Z, by finding the equations of...