.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by ...
.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by 3, and the even positive integers (b) R, and the interval (0, oo). (c) The interval [0,2), and the set [5,6)U7,8) (d) The intervals (-oo,-1) and (-1,0)
(a) Recall that two sets have the same cardinality if there is a bijection between them and that Z is the set of all integers. Give an example of a bijection f: Z+Z which is different from the identity function. (b) For the following sets A prove that A has the same cardinality as the positive integers Z+ i. A= {r eZ+By Z r = y²} ii. A=Z 1.
4. As we have seen, sometimes two sets can have the same cardinality even when one seems obviously much bigger than the other. Show that the following sets have the same cardinality. In part a, give a complete proof by finding a bijection. In part b, consider our proof that the rationals are countable. (a) The interval (0,1) and the real numbers, R (b) The integers, Z, and the Cartesian Product of the integers with itself, Zx Z
1. Show that if A and B are countable sets, then AUB is countable. 2. Show that if An are finite sets indexed by positive integers, then Un An is countable. 3. Show that if A and B are countable sets, then A x B is countable. 4. Show that any open set in R is a countable union of open intervals. 5. Show that any function on R can have at most countable many local maximals. Us
1)Complete each of the following statements using the words “greater than”, “less than” or “equal to” a) The cardinality of the even numbers is _________________ the cardinality of the natural numbers. b) The cardinality of the natural numbers is _________________ the cardinality of the positive rational numbers. c) The cardinality of the natural numbers is _________________ the cardinality of the rational numbers. d) The cardinality of the real numbers is _________________ the cardinality of the natural numbers. e) The cardinality...
Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties: (a) are divisible by 5 and by 7. (b) have distinct digits. (c) are not divisible by either 5 or 7.
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) Element R if and only if ad = bc. Show that R is an equivalence relation What is the equivalence class of of (1, 2), i.e. [(1, 2)]?
Eight consecutive three digit positive integers have the following property: each of them is divisible by its last digit. What is the sum of the digits of the smallest of the eight integers? A 10 B 11 С 12 D 13 E 14
1 For each of the following pairs of numbers a and b, calculate and find integers r and s such ged (a; b) by Eucledian algorithm that gcd(a; b) = ra + sb. ia= 203, b-91 ii a = 21, b=8 2 Prove that for n 2 1,2+2+2+2* +...+2 -2n+1 -2 3 Prove that Vn 2 1,8" -3 is divisible by 5. 4 Prove that + n(n+1) = nnīYn E N where N is the set of all positive integers....
3. Find the cardinality of the following sets. a. {x EZ: S10}. b. {€Z: € 0 c. 2211,2)