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5. Describe the following sets of real numbers and find the supremum and infimum of these...
1. Find the supremum and infimum of the following sets. (c) { (a) {, e} (b) (0,1) :n € N} (d) {r EQ : p2 <4} (e) [0, 1] nQ (f) {x2 : x € R} (8) N=1 (1 – 7,1+) (h) U-[2-7-1, 2”)
Can someone fully solve this for me please 5. (6 marks) For each of the following sets determine whether the supremum and infimum exist and if so, give the supremum and infimum. (You are not required to show any working for this question.) (a) Q (b) EN n+2 1,5 5. (6 marks) For each of the following sets determine whether the supremum and infimum exist and if so, give the supremum and infimum. (You are not required to show any...
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.
real analysis questions Find the interior of the following sets. (1): {1/n: neN}: (2): (0,5) (5, 7); (3): {re Q:0<r <2}. Classify each of the following sets as open, closed, or neither. (1): {: | - 51 < 1}; (2): {x: (x-3) > 1}; (3): {:13 -4)<4}.
For each of the following pairs of sets, prove that they are equinumerous. Remember that we have two ways to do this: we can find a bijection explicitly, or we can prove that there is an injection in each direction and then use the Schr¨oder-Bernstein theorem. 4. N and Qd for d > 1 5. R and R x R {a + bi |2 =-1, a,bE R} is the complex numbers) 6. R and C (where C
clear and clean answer,pls 2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There exists y in T such that for all x in S, y>x. Prove the following (of course, use the standard approach for proving universally and existentially quantified statements): (Q) For all x in S, there exists y in T such that y>x. 2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There...
Prove that the following relation R is an equivalence relation on the set of ordered pairs of real numbers. Describe the equivalence classes of R. (x, y)R(w, z) y-x2 = z-w2
2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There exists y in T such that for all x in S, y>x. Prove the following (of course, use the standard approach for proving universally and existentially quantified statements): (Q) For all x in S, there exists y in T such that y>x.
|х x.for some real numbers x and y. Find all ordered pars (r, y) such that yl and B (1 point) Suppose that A- AB BA. Enter your answer as a list of ordered pairs; for example, (1.2), (3,4), (5,6)) -1 Answer
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and y1 = 8 and for n = 1,2,3,··· x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y . nn nn (a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all positive integers n. (xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive integers n. Hence, prove...