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PROGRAM TITLE: HIGHER NATIONAL DIPLOMA IN COMPUTING
UNIT TITLE: UNIT 18 - DISCRETE MATHS
ASSIGNMENT NUMBER: 01
ASSIGNMENT NAME: SET THEORY AND FUNCTIONS
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TABLE OF FIGURES
Figure 1: Set and multiset example ........................................................................................ 6 Figure 2: Function example ................................................................................................. 10
PART 1:
1. Let A and B be two non-empty finite sets. If cardinalities of the sets A , B , and A ∩ B are 72, 28, and 13 respectively, find the cardinality of the set A ∪ B n ( A ∪ B ) = n( A ) + n( B ) – n( A ∩ B ) = 72 + 28 - 13 = 87 2. Let A = {n ∈ N: 20 ≤ n < 50} and B = {n ∈ N: 10 < n ≤ 30}. Suppose C is a set such that C ⊆ A and C ⊆ B****. What is the largest possible cardinality of C****? Since: C ⊆ A and C ⊆ B <=> C ⊆ A ∩ B => C = {n ∈ N | 20 ≤ n ≤ 30} <=> C ⊆ {20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} = 11 So, the largest possible cardinality of C is 11. 3. C onsider the sets A and B , where A = {3, | B |} and B = {1, | A |, | B |}. What are the sets? If | A| = 1 then B = {1, 1, | B |} and A = {3, | B | = 3} B = {1, 1, | B |} <=> B = {1, | B |} <=> B = {1, 3} => | B | = 2 => Wrong! If | A | = 2 then B = {1, 2, | B| } and A = {3, | B| ≠ 3 } and |B| < 3
import import java.util.ArrayListjava.util.HashSet; ; import import java.util.java.util. ListSet ;; public class public static void Main { main(String[] args){ // Using Set Set companies = new HashSet<>(); companies companies..addadd(("Netflix""Tesla");); companies companies..addadd(("Softbank""Apple"); ); companies System.out.add.println("American Express"(companies.toString); ()); // Using Multiset List peopleName = new ArrayList<>(); peopleName peopleName..addadd(("John""Gary");); peopleName peopleName..addadd(("John""Margaret"); ); peopleName System.out..addprintln("Gary"(peopleName); .toString()); }^ }
And here we have the result, as you can see the names John and Gary can be duplicated on the multiset.
Figure 1 : Set and multiset example
PART 3:
1. Determine whether the following functions are invertible or not. If it is invertible, then find the rule of the inverse f-1 (x)
i. f (x) = x^2 If x = 2, then f(x) = 4 If x = - 2 then f(x) = 4 From two statements above => The function f(x) = x^2 is not one-to-one. So in conclusion, the function is not invertible
ii. f(x) = (^) 𝑥^1
Let y = (^1) 𝑥 with y ≠ 0 <=> x = (^1) 𝑦 <=> x × y = (^1) 𝑦 × y <=> x × y = 1 <=> y = (^1) 𝑥 So the function is invertible and f-1(x) = (^) 𝑥^1
2. Function f(x) = 𝟓 𝟗 (x-32) converts Fahrenheit temperatures into Celsius. What is the function for opposite conversion? Let y = 59 (x-32) <=> 9y = 5x – 160 <=> 9y + 160 = 5x <=> x = 95 y + 32 So the opposite function, which converts Celsius to Fahrenheit is 95 x + 32 where x is the temperature in Celsius. 3. Present the application of function in software engineering? Give a specific programming example The main purpose of any function is to take an input and return the correct output. So by using functions, software developers can make their programs perform tasks from simple to extremely complex involving complicated logic ones. Here is an example of using a function to convert Mile to Kilometer in Java public class final static double Main { MI_TO_KM_CONSTANT = 1.6093; public static void main(String[] args) throws InterruptedException { System System..outout..printlnprintln(("1 mile to km: ""10 miles to km: " + convertMileToKm + convertMileToKm ( 1 ));( 10 )); System System..outout..printlnprintln(("25 miles to km: ""178 mile to km: " ++ convertMileToKmconvertMileToKm (( 25178 ));)); } public static double return mile * MI_TO_KM_CONSTANT convertMileToKm;(double mile){ }^ } And here are the result:
Figure 2 : Function example
