# MAT235 Course Introduction

at University of Calgary (Calgary, Alberta)

Learn Discrete Mathematics II (MAT235) in the online version of University of Calgary.

The course introduces students to the concepts and techniques of discrete mathematics. Topics include sets, functions, relations, logic, proof techniques, recursion and automata. After covering sets and functions, we introduce a variety of basic graph theory problems.

The course assumes a basic understanding of the following: sets; functions; matrix representations; algebraic structures; equivalence relations; Boolean

# MAT235 Course Description

Discrete Mathematics II. Continuation of topics covered in MAT235. Prerequisites: MAT 135 or equivalent.

MAT235L Course Description for MAT235L – Discrete Mathematics II (MAT235L) Discrete Mathematics II. Continuation of topics covered in MAT235.

MAT235U Course Description for MAT235U – Discrete Mathematics II (MAT235U) Discrete Mathematics II. Continuation of topics covered in MAT235.

MAT245 Course Description for MAT245 – Discrete Structures

# Universities Offering the MAT235 Course

Universities Offering the MAT235 Course for MAT235 – Discrete Mathematics II (MAT235) Colleges offering the MAT235 Course for MAT235 – Discrete Mathematics II (MAT235)

# MAT235 Course Outline

– Fall 2008

Course Description

Discrete Mathematics II (MAT235) is a graduate course in discrete mathematics. It is designed to be the continuation of MAT235 and MAT235A. The objectives of MAT235 are to:

develop an understanding of the foundations of mathematical structures; provide examples, exercises, and problems that require the use of these structures;

introduce some basic algebraic concepts such as powers, polynomials, exponents, logarithms and sequences; and

provide students with

# MAT235 Course Objectives

The goal of this course is to develop a solid foundation in Discrete Mathematics. This will be accomplished by studying a subset of topics from Analysis, Combinatorics, Probability Theory and other related fields.1) Discrete Mathematics Topics1. Linear Algebra: matrices, determinants, vectors, eigenvalues2. Introduction to Probability Theory3. Graphs and Trees4. Sets5. Relations6. Functions7. Continuity8. First-Order Logic9. Counting10.Determinant

# MAT235 Course Pre-requisites

(0.5) A student must have taken MAT235 Discrete Mathematics I (MAT235) (0.5) before MAT235 Discrete Mathematics II (MAT235). Course Content The course content of MAT235 is based on topics from the Discrete Mathematics I course (see below). The course involves the following topics: Introduction to Discrete Mathematics, Linear Algebra, Boolean Algebra, Quantifiers and Inequalities, Logic and Proof, Functions, Relations and Functions, Graphs, Decision Diagram

# MAT235 Course Duration & Credits

Course Description Discrete Mathematics II is a continuation of Discrete Mathematics I. Topics covered in the course include definitions and formal proofs, recursive relations, computability, recursion, language theory and intuitionistic logic. There will be an emphasis on applications of discrete mathematics to computer science. Prerequisite: MAT235 and MAT261C or consent of instructor . Credits: 3 Instructor(s): Lea Buell Class Notes / Online Textbook:

Click Here

MAT235 Course Detail

Discrete Math II (

# MAT235 Course Learning Outcomes

These are the Course Learning Outcomes for MAT235 Discrete Mathematics II (MAT235) listed on the Discrete Mathematics home page at http://www.cs.ucla.edu/~mathe/mat235.html.

MAT235 Syllabus for Discrete Mathematics II (MAT235) Here is a description of the course material. You may find that this looks too high level or abstract. You are welcome to ask questions about it, and I will be happy to clarify it.

Discrete Mathematics II (

# MAT235 Course Assessment & Grading Criteria

Page 2 of 6

3 Grade Distribution and Percentile Rank Distinctive Features Total % A Excellent B Good C Moderate D Fair E Pass F Fail G Disserviceable I Incomplete Notes: The appropriate grade for each item will be determined by applying the following rules: a. If the student obtained a score of three or higher, it is a pass grade. b. If the student obtained a score between three and nine, it is an incomplete grade. c. If the student obtained

# MAT235 Course Fact Sheet

Course Credit Hours: 3 (3 Lecture, 0 Lab) Prerequisite: MAT235 or consent of instructor Discrete Mathematics II is a continuation of MAT235. This course focuses on the mathematics of sets, functions, and discrete structures and their applications to computer science. It requires no prior knowledge of finite math, but a working knowledge of set theory and elementary calculus would be helpful. Homework assignments will usually be based on questions that arise in lecture or during discussion sections.

MAT235 Course Fact Sheet

# MAT235 Course Delivery Modes

Discrete Mathematics II (MAT235)

Discrete Mathematics II Course Content Discrete Mathematics II Course Overview

Mathematics and Codes

Vectors and Matrices

Graph Theory

Sequence and Series

Counting and Probability

Boolean Algebra and Boolean Functions

Duality Method in Linear Algebra with Applications to Graph Theory

Matrices, Determinants, Linear Transformations, Eigenvalues, Eigenvectors in R^2

Multi-Dimensional Duality Methods in Linear Algebra with Applications to Graph Theory

# MAT235 Course Faculty Qualifications

Course Description The second course in the Discrete Mathematics sequence, MAT235 introduces students to higher level mathematics that is relevant to the area of discrete mathematics. The main emphasis of this course is on the ability to understand and analyze mathematical problems and their solutions, rather than on memorizing mathematical rules. This course is designed to enable students to comprehend and work with proofs as well as developing skills needed for analyzing problems from various perspectives. The emphasis of MAT235 will be on the development of a strong background in the

# MAT235 Course Syllabus

This course introduces students to various mathematical concepts that are required to understand the other topics in discrete mathematics. The course also offers a review of the topics introduced in Discrete Mathematics I (MAT235), and includes examples that introduce students to Boolean algebra, sets, graphs, functions, sequences, and matrices. This course also provides an introduction to the study of proofs using deductive reasoning and informal logic. The teaching methodology is bottom-up: theory builds upon prior knowledge while exercises are designed as problem solving exercises.

# Suggested MAT235 Course Resources/Books

Name Course Book Author ISBN GLEITZMANN 6th ed. Discrete Mathematics and Its Applications

GEM 5th ed. Discrete Mathematics Kleinberg and Tardos 5th ed. Fundamentals of Discrete Mathematics

Nakamura, S., and Shaffer, C., editors. Discrete Mathematics and Its Applications, Second Edition McGraw-Hill/Irwin, Upper Saddle River, NJ

# MAT235 Course Practicum Journal

– Week 3 (Week 3, First Quarter, Fall 2009) Part B—Problem Set This problem set is due on Wednesday, September 23, 2009. Problem Set #2 —Consider the following two vectors: • r1 = (4, -2); • r2 = (-5,-3). These are both non-zero vectors with coordinates {4,-2} and {(-5,-3)}. Compute each of the following: (a) |r1

# Suggested MAT235 Course Resources (Websites, Books, Journal Articles, etc.)

1. Discrete Mathematics II (MAT235) – Spring 2016

2. Discrete Mathematics II (MAT235) – Fall 2015

3. Discrete Mathematics II (MAT235) – Summer 2015

4. Discrete Mathematics II (MAT235) – Spring 2015

5. Discrete Mathematics II (MAT235) – Fall 2014

6. Discrete Mathematics II (MAT235) – Spring 2014

7. Dis

# MAT235 Course Project Proposal

– Page 1 of 7 Discrete Mathematics II Course Project Proposal (Please Read and Answer all Questions) The following course project is required for MAT235. It is due at the beginning of the semester. Please read this entire document carefully before submitting your proposal. This project must be done on your own time and you are responsible for making sure that it will be submitted by the due date.

Mathematics – Software Engineering

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# MAT235 Course Practicum

List of Topics Time Topic Sections Day 1 Thursday, February 14, 2019 Course Introduction General Concepts Linear Algebra Matrix Operations Linear Functions and Graphs Exponential and Logarithmic Functions Boolean Algebra Sets Operations on sets Basic Counting Problems and Sorting Routines Classes of Sets Complements and Set Operations Solving Geometry Word Problems Proofs in Geometry Collatz Conjecture Integers Sets of Integers (infinite) Properties of Sets (including Pigeonhole Principle) Systematic Assignment 2

# Related MAT235 Courses

MAT235 Discrete Mathematics II 4.0 Credits This course is an introduction to the topic of mathematical logic with a focus on logic in mathematics. Readings will be from seminal works in this field, such as Harris’ “The Model Theory of Natural Deduction” and Brouwer’s “Logic”. Students will also be introduced to the formal semantics and verification techniques necessary for proving that propositions can be proved using the rules of arithmetic. Course Objectives: The student should be able to prove propositions

# Midterm Exam

— 2018

Review of Linear Algebra and its Applications to Mathematics (MAT245)

## Top 100 AI-Generated Questions

by Andrew A. Martin

This is a list of 100 questions for MAT235 Discrete Mathematics II – Continuous Mathematics III (MAT235) at the University of Wisconsin-Madison in 2016. The questions are generated by using the book’s text and exercises as a base and randomly rewording them, generating new questions and updating the book to reflect these changes. The generated questions can be found in the second section of the book titled “Inquire,” where they will be used to

## What Should Students Expect to Be Tested from MAT235 Midterm Exam

Midterm Exam (Midterm) – Discrete Mathematics II (MAT235) Midterm Exam (Midterm) – Discrete Mathematics II (MAT235) Midterm Exam (Midterm) – Discrete Mathematics II (MAT235) Midterm Exam (Midterm) –

M: I. Gauss-Markov Theorem. 2, 4: II. Matrix Optimization. 6, 8, 10, 12: III. Graphs and Trees.

I.B.

## How to Prepare for MAT235 Midterm Exam

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## Midterm Exam Questions Generated from Top 100 Pages on Bing

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# Final Exam

at University of Waterloo from May 29, 2018 to June 3, 2018. If you are taking this course as an elective and do not plan to take the exam, skip section 11.2 (f) below. If you are taking this course for a grade in this course and want to take the exam, skip section 11.2 (f) and read sections 11.2 (g), 11.2 (h), and 11.2

## Top 100 AI-Generated Questions

– Spring 2018 | Udemy

Course Summary

Here is a list of the top questions for MAT235 – Discrete Mathematics II (MAT235) at Universidad de las Américas Puebla.

Here are the top 100 most asked questions from students enrolled in MAT235 – Discrete Mathematics II (MAT235).

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Course Topics and Objectives

This course introduces the candidate to mathematical models of

## What Should Students Expect to Be Tested from MAT235 Final Exam

? This is the Discrete Mathematics II (MAT235) Final Exam for MAT235 – Discrete Mathematics II (MAT235). You can find this exam in the chapter Discrete Mathematics II from the textbook. The examination consists of multiple choice questions. You should know the answers to all questions on this exam because it is critical to your grade in the course. It is highly recommended that you take a study course before taking this final exam because it will help you learn what to expect from the exam and

## How to Prepare for MAT235 Final Exam

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## Final Exam Questions Generated from Top 100 Pages on Bing

## Final Exam Questions Generated from Top 100 Pages on Google

– A Term Paper submitted by Adrian Constantinescu

MAT 235 – Discrete Mathematics II (Mat 235) – A Term Paper 1. What is a set? Is a set necessarily finite or infinite? Explain why you think this. A set is an unordered collection of objects in which the members are distinct but not necessarily equal to each other. Thus, a set can be infinite or finite, but it is always one of these two states. The term “set” actually means the

# Week by Week Course Overview

## MAT235 Week 1 Description

MAT235 Week 1 Description for MAT235 – Discrete Mathematics II (MAT235) DISCRETE MATHEMATICS II * For exam pl

WEEK 2 INSTRUCTIONS FOR WEEK TWO DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING MATS 226F PROJECT #4 Introduction to Programming Using Visual Basic .NETThe name of the project is “Introduction to Programming Using Visual Basic .NET”. You are required to read and understand all sections of this project, including Chapter 8, “

## MAT235 Week 1 Outline

– Homework Help

A comparison of the two discrete mathematics courses, MAT235 and MAT137. What are the differences? What is covered in each course?

MAT235 Week 1 Assignment: Discrete Mathematics II (Discrete Math) (MA…

MAT235 Assignment: Discrete Mathematics II Due Date: Day 7 [link] This assignment introduces a discrete math course. Students will be introduced to concepts and algorithms needed to solve discrete math problems.

MAT235 Assignment | MAT235 at UOP Course

## MAT235 Week 1 Objectives

Week 1 Objectives for MAT235 – Discrete Mathematics II (MAT235) Click on the link below to purchase a+ score http://www.homeworkmade.com/product/… For more classes visit www.homeworkmade.com MAT235 Week 1 Objectives for MAT235 Discrete Mathematics II (MAT235)Click on the link below to purchase a+ score For more classes visit www.homeworkmade.com

MAT-235 Week 4 DMS Question and Answer.pdf

Description

## MAT235 Week 1 Pre-requisites

– Duration: 3:13. The course is taught in session mode and is a graduate level course. MAT235 Discrete Mathematics II (Spring 2015) – University of Toronto Home Page Introduction to Matrix Algebra and Determinants (MAT235) – Spring 2018.

Matrices are an integral part of mathematics, but I remember struggling through some matrices at school. If you’re studying this at university, it’s time to get acquainted with them all over again – from the basic operations

## MAT235 Week 1 Duration

University of Phoenix Material MAT235 Week 1 Individual Assignment Math in the Real World Table of Contents Description 3 Description 3 Format and Structure 4 Purpose 5 Developmental Goals for this Course 5 Learning Objectives 6 The Academic Success Center 6 Assignment Guidelines 6 Appendix A Worksheet: Positive and Negative Examples from Class (MAT235) 8 Appendix B Sample Problem Set (MAT235) MAT235 Week2 for the Real World (MAT235) Part I: Matrices, In

## MAT235 Week 1 Learning Outcomes

– College Homework And …

Introduction to Discrete Mathematics – College Homework Help and Online Tutoring

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Discrete Mathematics II MAT235 Week 1 Homework Assignment

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MAT235 Week 2 Learning Outcomes for MAT235 – Discrete Mathematics II (MAT235) – College Homework And …

MAT235 Week 2 Learning Outcomes

## MAT235 Week 1 Assessment & Grading

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## MAT235 Week 1 Suggested Resources/Books

MAT235 Week 1 Suggested Resources/Books for MAT235 – Discrete Mathematics II (MAT235) 5 Pages 1392 Words December 2014 Download this PDF file Click to download Embed

Discrete Mathematics: The Basis of Computational Thinking (2013) By Thomas C. Hobbs, Ph.D.

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## MAT235 Week 1 Assignment (20 Questions)

(MAT235 Week 1 Assignment) for DISCRETE MATHEMATICS II (MAT235) Discrete Mathematics II (MAT235) • This assignment is worth 20 points and is due on Friday. If you have any questions please contact us at [email protected] Use your own words in this posting. Assignment Guide for MAT232-001 Discrete Mathematics I: Study Guide I: Recall S Tudy Guide to help prepare for the lecture and textbook assignments. The study guide contains everything

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## MAT235 Week 1 Discussion 1 (20 Questions)

Week 1 Discussion 1 (20 Questions) for MAT235 – Discrete Mathematics II (MAT235) In this discussion, you will learn how to: • Identify and explain the structure of a discrete mathematics course. Read More

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## MAT235 Week 1 DQ 1 (20 Questions)

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## MAT235 Week 1 Discussion 2 (20 Questions)

Week 1 Discussion 2 (20 Questions) for MAT235 – Discrete Mathematics II (MAT235) Question No. 1 Which of the following is NOT a subset of a set A? a. A = {3,4,5,6} b. A = {2,4} c. A = {(3,4), (5)} d. All of the above are subsets of A. Question No. 2 If an … Read More

## MAT235 Week 1 DQ 2 (20 Questions)

You are required to answer 20 questions (80 points) regarding the problem: Answer the questions in

Here is a list of all the possible answers for MAT235 Week 1 DQ 2 (20 Questions): Question #1: (1 point) Is there an optimal implementation for each of the following problems? Let us consider two

MAT235 Week 4 DQ 2 (15 Points) MatLab is a general-purpose programming language. It has some features which make it better suited

## MAT235 Week 1 Quiz (20 Questions)

Week 1 Quiz (20 Questions) for MAT235 – Discrete Mathematics II (MAT235) Questions and Answers.

Learn: MAT235 MAT235 Week 3 Quiz Part 2 (20 Questions) for MAT235 – Discrete Mathematics II (MAT235) Week 3 Quiz Part 2 (20 Questions). Question: The following are all valid ways to represent the square root of a number. If you subtract each digit from the previous digit, you will get the same result. What is

## MAT235 Week 1 MCQ’s (20 Multiple Choice Questions)

at Southern New Hampshire University (SNHU) – Page 1

SCENARIO 1: A group of six students is studying the following discrete mathematics topics:

a. Probability

b. Discrete mathematics

c. Graph theory

d. Graph drawing

e. Decision-making theory

In this scenario, you are responsible for creating a questionnaire that contains 20 multiple choice questions.

You have an initial idea for question a), and you want to make it more rigorous and include more data analysis,

## MAT235 Week 2 Description

In this week’s lesson, you will learn to analyze a problem that involves two or more variables. You will find out how to determine if a given set of data is discrete or continuous. You will also be able to use the software package Excel to work on problems involving graphs and to analyze functions. [Read More]

## MAT235 Week 2 Outline

Discrete Math I Semester Fall 2016 Outline for MAT235 – Discrete Mathematics II (MAT235) Discrete Math I Semester Fall 2016

Week 2 Outline for MAT235 – Discrete Mathematics II (MAT235) Discrete Math I Semester Fall 2016 Week 2 Outline for MAT235 – Discrete Mathematics II (MAT235) Discrete Math I Semester Fall 2016

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## MAT235 Week 2 Objectives

– ECE225 – Introduction to Electromagnetic Theory (ECE225) – MAT235 – Discrete Mathematics II (MAT235) – 1 Assignment 1

Assignment 1 2 Write a C program that accepts an integer n and prints the square of the number.

Write a C program that accepts an integer n and prints the square of the number. Read a character from standard input, and parse it according to rules for characters in the ASCII character set.

If your program uses file

## MAT235 Week 2 Pre-requisites

You are required to know the following: • MAT235 Pre-requisites (MAT235) – Discrete Mathematics II (MAT235) 1. The Cartesian product of two sets is a new set, and the union of two sets is a new set. This function accepts an integer value and returns a new integer value between 0 and 1. For example, if you have the following data: [16, 5, 9, 4], the function will return a number that

## MAT235 Week 2 Duration

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## MAT235 Week 2 Learning Outcomes

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## MAT235 Week 2 Assessment & Grading

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Mat 235 week 1 assignment 1 answer key

What type of forces is the mass-moment diagram for? the “force of gravity” If a $9.00 bill were deposited in a jar at a rate of $3.00 per hour for 15 hours each day, calculate how much $10 bill would have grown by the end of its life at that rate

## MAT235 Week 2 Suggested Resources/Books

Week 2 Suggested Resources/Books for MAT235 Discrete Mathematics II Courseware: – Complete the following week-by-week assignments for DISCRETE MATHEMATICS II. Write an essay of approximately 800-1,000 words for each assignment. This includes the entire course, as well as every assignment on

INSTRUCTIONS – MatLab for Data Analysis and Simulation I are found in chapter 1 of the textbook. The codes are also available at http://www.cs.iit.edu/~

## MAT235 Week 2 Assignment (20 Questions)

to pass this course.

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## MAT235 Week 2 Assignment Question (20 Questions)

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## MAT235 Week 2 Discussion 1 (20 Questions)

for

MAT 235 Discrete Mathematics II Assignment #1 (20 Questions) For this assignment, you will be required to do some work in the area of graph theory. Specifically, you will be asked to write a paper analyzing two well-known graph classes: the digraph and planar graphs. Your paper should include a reference list as well as the following parts: • A brief discussion of what each of the graph classes is (digraph, planar graphs). What are the main differences between

## MAT235 Week 2 DQ 1 (20 Questions)

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## MAT235 Week 2 Discussion 2 (20 Questions)

for Free. Discrete Mathematics II – Critical Thinking and Problem Solving with the DVD (MAT235) for Free The following items are included in this course: 1 DVD set of lecture videos on discrete mathematics and problem solving, complete PowerPoint lectures with animations and video clips, quizzes, evaluation sheet, midterm exam. HINT: For this quiz you must know your class time to be able to use the open book policy correctly.

Chapter 5 – Basic Probability Concepts and Odds. Chapter 6 –

## MAT235 Week 2 DQ 2 (20 Questions)

Week 2 DQ 2 (20 Questions) for MAT235 – Discrete Mathematics II (MAT235) Week 2 DQ 2 (20 Questions) for MAT235 – Discrete Mathematics II (MAT235) Week 2 DQ 2 (20 Questions) for MAT235 – Discrete Mathematics II (MAT235) Week 2 DQ 2 (20 Questions) for MAT235 – Discrete Mathematics II (MAT235)

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## MAT235 Week 2 Quiz (20 Questions)

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## MAT235 Week 2 MCQ’s (20 Multiple Choice Questions)

in the subject MAT235 at University of Florida

Question 1

An integer is stored on a 64-bit computer as:

A. A word

B. An integer word

C. A byte

D. None of the above

Question 2

If M contains integers n and n-1, then the smallest number that can be stored in M such that m+n=m is:

A. N + 1

B. N + 2

C. n+1 +

## MAT235 Week 3 Description

– Midterm Exam | SIU SOLIDWORKS Simulation Reviews and Training MAT235 – Discrete Mathematics II (MAT235) – Midterm Exam | SIU SOLIDWORKS Simulation Reviews and Training Matheo Maldonado, 2017-05-08 09:43:47. Math 254 Matheo Maldonado 2014-04-15 – MIDTERM EXAM PRACTICE – PHY225/CS226 SUMMER ASSIGNMENT 2

## MAT235 Week 3 Outline

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I will use this assignment to teach you how to write a basic research proposal, as well as how to write an experiment with the null hypothesis and experiment design. First we’ll start by learning what a research question is, and then we’ll move on to learning how to formulate your own research questions. Finally, we’ll learn about the three types of hypotheses: two-way (Null Hypothesis), one-way (Alternative Hypothesis) and mixed hypotheses.

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## MAT235 Week 3 Objectives

– NIOSHS. 4 Week 3 Quiz 1 (10 pts) 5 Week 3 Quiz 2 (10 pts) 6 Homework #4 due on September 14. MAT235 course materials can be found at:. This course is the first of three courses in Discrete Mathematics. NET Framework Version 4. • Unit II: Set Theory, Probability and Statistics, Relational Calculus, Logic and Model Theory • Unit III: Function Spaces, Banach Spaces and Operators

## MAT235 Week 3 Pre-requisites

Discrete Mathematics II (MAT235) View the complete course: http://ocw.mit.edu/6-201 Discrete Math via Real-World Applications 6.1 Integers, integers modulo and powers of 2, naturals, whole numbers, empty set and set theory. 6.2 Sets, including intersection of sets, unions of sets, intersection and union of sets with the empty set. 6.3 Properties of subsets: complementation and intersection formulae. 6

## MAT235 Week 3 Duration

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Lecture Notes on Discrete Mathematics: Matrices and Graphs (II) Lecture Notes on Discrete Mathematics: Matrices and Graphs (II) This lecture will deal with matricies, graphs and their applications. It will also cover the basic concepts of graph theory. The class is to be conducted in English. Beforehand the students should know how to solve linear equations in two variables, how to write

## MAT235 Week 3 Learning Outcomes

Week 3 Assignment 1: Linear Equations in Two Variables

Question 1. Given the following graph of the number of neighbors for two points A and B on a graph and you know that A has a distance of 5 from B and B has a distance of 7 from A, calculate the length of AB using one variable.

Question 2. Find an equation for the line passing through the point (5,6) and is parallel to y = -1/2x +

## MAT235 Week 3 Assessment & Grading

(Grading for MAT235 Week 3 Assessment & Grading for MAT235 – Discrete Mathematics II (MAT235) (Grading for MAT235 Week 3 Assessment & Grading for MAT235 – Discrete Mathematics II (MAT235) (Grading for MAT235 Week 3 Assessment & Grading for MAT235 – Discrete Mathematics II (MAT235) Assignment

MAT205

Instructor: March 2016

E-mail: Dr. Debojyoti Dey

## MAT235 Week 3 Suggested Resources/Books

Week 3 Suggested Resources/Books for MAT235 Discrete Mathematics II (MAT235) You will need to get your own copy of the textbook in this course. Some of the links below are available for free through the OER Commons, an open source repository. http://oercommons.org/ https://www.ebrary.com/science/discrete-mathematics-ii-mat235/book/11009101/chapter/16112147 The following textbooks are also available through ebr

## MAT235 Week 3 Assignment (20 Questions)

– Course Hero

MAT235 week 3 assignment (20 questions)

Question 1

A characteristic function of a discrete random variable is the probability density function.

(a) Find the probability that a randomly selected real number is less than or equal