Question 5. Let x,y E R. Prove that if x and y are irrational, then at...
# 7 please 6. Prove that if x is rational and y is irrational, then 2 +y is irrational. 7. Prove that if x, y € R+ such that Ty # #4, then x + y.
Let x,y ∈ R. Which of the following statements are true. If the statement is true prove it, if not give a counterexample a) If x is rational and y is irrational, then x y is irrational. b) If x and y are both irrational then x + y is irrational. c) Ifx and y are both irrational then ry is irrational d) Ifx is rational and y is irrational then ry is irrational.
Let x,y,z e Z. Prove that if x+y= 2, then at least one of , y, and z must be even.
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.) Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2 {(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2
5. Let A =R x R and f: A+ A be given by the rule f(x, y) = (x – y, x + y). (a) Prove f is one-to-one. (b) Prove f is onto A. (Comment: don't forget that if given b E A, you construct a such that f(a) = b, you must also show a E A.) (c) What is the inverse function? (d) Is f a permutation? Explain.
11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
Let X, Y E Mn (R). Prove that XY = XY_if and only if there exists an invertible matrix Z so that X = Z In and Y = Z1 + In. Hint: the trace is not involve at all in this problem _
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...