(b) Uniqueness of multiplicative inverse. Prove: If y E R is any real number with the...
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
7. Prove that for any positive real number r, if r is not an integer, then [x]+-1= 1
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
9. Let x,y > 0 be real numbers and q, r E Q. Prove the following: (а) 29 > 0. 2"а" and (29)" (b) x7+r (с) г а — 1/29. 0, then x> y if and only if r4 > y (d) If q (e) For 1, r4 > x" if and only if q > r. For x < 1, x4 > x* if and only if q < r.
(6) Let a be a positive real number. Note that for all r, y R there exists a unique k E Z and a unique 0 Sr <a so that Denote (0. a), the half open interval, by Ra and define the following "addi- tion" on Rg. where r yr + ka and r e lo,a) (a) Show that (Ra. +a) is a group (b) Show that (Ri-+i ) İs isomorphic to (R, , +a) for any" > O. (Therefore,...
Question 8: For any integer n 20 and any real number x with 0<<1, define the function (Using the ratio test from calculus, it can be shown that this infinite series converges for any fixed integer n.) Determine a closed form expression for Fo(x). (You may use any result that was proven in class.) Let n 21 be an integer and let r be a real number with 0<< 1. Prove that 'n-1(2), n where 1 denotes the derivative of...
Prove that there is a unique ordered pair (x, y) where x and y are real numbers such that y=x’ and y=2x-1. (Be sure to prove both “existence" and "uniqueness.") (25 pts)
.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...
Prove that for any real number x > 0,