3. Suppose A and B are non-empty sets of real numbers that are
both bounded above.
(a) Prove that, if A ⊆ B, then supA ≤ supB.
(b) Prove that supA∪B = max{supA,supB}.
(c) Prove that, if A∩B 6= ∅, then supA∩B ≤ min{supA,supB}. Give
an example to show that equality need not hold.
A, B and C be sets.
(a) Suppose that A ⊆ B and B ⊆ C. Does this mean that A ⊆ C?
Prove your answer. Hint: to prove that A ⊆ C you must prove the
implication, “for all x, if x ∈ A then x ∈ C.”
(b) Suppose that A ∈ B and B ∈ C. Does this mean that A ∈ C?
Give an example to prove that this does NOT always happen (and
explain why...
Let A and B be sets of real numbers such that A ⊂ B. Find a
relation among inf A, inf B, sup A, and sup B.
Let A and B be sets of real numbers and write C = A ∪ B. Find a
relation among sup A, sup B, and sup C.
Let A and B be sets of real numbers and write C = A ∩ B. Find a
relation among sup A, sup B, and sup...
1. Suppose we have two sets of prediction for the inflation rate
for next year: one from a random sample of Fortune 500 firms and
another from a random sample of university economists. At 0.05
level can it be claimed that on the average university educators
are predicting a higher inflation rate for next year than the major
private businesses?
Fortune 500 Firms
4.3%
3.8%
6.0%
4.4%
5.1%
5.6%
4.2%
6.1%
4.5%
4.0%
University Economists
4.4%
5.9%
7.0%
5.1%
5.9%...
goal Find the average of the elements in the list (list could
have any number of elements).
If the average is a decimal number, return only the integer part.
Example: if average=10.8, return 10
Example: if list={10, 20, 30, 40, 50}, the method should return:
30
import java.io.*;
import java.util.Scanner;
class Node {
int data;
Node next;
Node(int d){ // Constructor
data = d;
next = null;
}
}
class LinkedList {// a Singly Linked List
Node...
You have n numbers and you want to find a number x (out of the
n) such that x is larger than the median. You can create an
algorithim that takes time O(nlogn): sort the n numbers and then
report any number that is larger than the element in position n2 of
the sorted array. You can also create an algo in O(n) time, by
finding the median in linear time and then doing a linear scan to
find a...
1. (14) List the elements for each of the following sets:
(1) P({a, b, c}) (Note: P refers to power set)
(2) P{a, b}) - P({a, c})
(3) P(Æ)
(4) {x Î ℕ: (x £ 7 Ù x ³ 7} (Note: ℕ is the set of nonnegative integers)
(5) {x Î ℕ: $y Î ℕ (y < 10 Ù (y + 2 = x))}
(6) {x Î ℕ: $y Î ℕ ($z Î ℕ ((x = y +...
prove or disppprove.
Suppose A & B are sets.
(1) A function f has an inverse iff f is a bijection.
(2) An injective function f:A->A is surjective.
(3) The composition of bijections is a bijection.