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Chapter 1: Problem 16

A discrete mathematics class contains 1 mathematics major who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores, 2 mathematics majors who are juniors, 2 computer science majors who are juniors, and 1 computer science major who is a senior. Express each of these statements in terms of quantifiers and then determine its truth value. a) There is a student in the class who is a junior. b) Every student in the class is a computer science major. c) There is a student in the class who is neither a mathematics major nor a junior. d) Every student in the class is either a sophomore or a computer science major. e) There is a major such that there is a student in the class in every year of study with that major.

Short Answer

Expert verified
a) True, b) False, c) True, d) False, e) False

Step by step solution

01

Title - Quantify the given class composition

Break down the total number and major/year combination of students in the class: 1 freshman mathematics major, 12 sophomore mathematics majors, 15 sophomore computer science majors, 2 junior mathematics majors, 2 junior computer science majors, 1 senior computer science major.
02

Title - Statement a) There is a student in the class who is a junior

Express this in quantifiers: \( \exists x (Junior(x)) \). Since there are 2 mathematics juniors and 2 computer science juniors, the statement is True.
03

Title - Statement b) Every student in the class is a computer science major

Express this in quantifiers: \( \forall x (Student(x) \rightarrow CS(x)) \). Since there are mathematics majors in the class, the statement is False.
04

Title - Statement c) There is a student in the class who is neither a mathematics major nor a junior

Express this in quantifiers: \( \exists x (eg MathMajor(x) \land eg Junior(x)) \). There are computer science freshmen, sophomores, and seniors, making this statement True.
05

Title - Statement d) Every student in the class is either a sophomore or a computer science major

Express this in quantifiers: \( \forall x (Student(x) \rightarrow (Sophomore(x) \lor CS(x))) \). There are juniors in various majors, making this statement False.
06

Title - Statement e) There is a major such that there is a student in the class in every year of study with that major

Express this in quantifiers: \( \exists m (\forall y (Year(y) \rightarrow \exists x (Student(x) \land Major(x,m) \land YearOfStudy(x,y)))) \). Since no single major covers all four years of study (freshman, sophomore, junior, senior), the statement is False.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantifiers
In discrete mathematics, quantifiers are symbols used to express the number of elements within a specified set that satisfy a given property. The two main types of quantifiers are:
  • **Existential Quantifier (\(\exists\))**: Indicates that there exists at least one element in a set for which a property holds. For example, \(\exists x (Junior(x))\) means 'there exists an x such that x is a junior.'
  • **Universal Quantifier (\(\forall\))**: Indicates that a property holds for all elements in a set. For instance, \(\forall x (Student(x) \rightarrow CS(x))\) translates to 'for all x, if x is a student, then x is a computer science major.'
By using these quantifiers, we can create precise mathematical statements about groups of objects, like the students in our classroom scenario, and examine their respective properties efficiently.
Truth Value
The truth value of a logical statement determines whether the statement is true or false. When you express statements using quantifiers, you determine their truth by examining the elements in the set. For example:
  • **Statement (a)**: \(\exists x (Junior(x))\) asserts that there is at least one student who is a junior. In our classroom, there are 2 mathematics juniors and 2 computer science juniors. Therefore, the statement is **True**.
  • **Statement (b)**: \(\forall x (Student(x) \rightarrow CS(x))\) states that every student is a computer science major. Given that many students are mathematics majors, this statement is **False**.
Determining the truth value involves checking the conditions posed by the quantifiers and verifying if the elements meet these conditions.
Student Classification
Classifying students based on major and year of study helps simplify the analysis of logical statements. Here's a breakdown of our class composition:

  • 1 freshman mathematics major
  • 12 sophomore mathematics majors
  • 15 sophomore computer science majors
  • 2 junior mathematics majors
  • 2 junior computer science majors
  • 1 senior computer science major

This classification enables us to understand the group's diversity and aids in the formation and verification of logical statements. For example, it's clear from our classification that not every student is a computer science major, making statement (b) false.
Logical Statements
Logical statements in discrete mathematics use variables, quantifiers, and logical connectives (like AND, OR, NOT) to describe properties and relationships within a set. Here's how we express and analyze them:
  • **AND (\(\land\))**: The statement A AND B (\(A \land B\)) is true only if both A and B are true. For example, \(\exists x (eg MathMajor(x) \land eg Junior(x))\) means there exists a student who is neither a mathematics major nor a junior.
  • **OR (\(\lor\))**: The statement A OR B (\(A \lor B\)) is true if at least one of A or B is true. For instance, \(\forall x (Student(x) \rightarrow (Sophomore(x) \lor CS(x)))\) means every student is either a sophomore or a computer science major.
Understanding and accurately forming these logical statements is key to exploring the relationships and properties within groups, such as students in our classroom scenario.

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Most popular questions from this chapter

Prove that there are infinitely many solutions in positive integers \(x, y,\) and \(z\) to the equation \(x^{2}+y^{2}=\) \(z^{2} .\left[\text { Hint: Let } x=m^{2}-n^{2}, y=2 m n, \text { and } z=m^{2}+n^{2}\right.\) where \(m\) and \(n\) are integers. \(]\)

The Logic Problem, taken from \(W F F^{\prime} N\) PROOF, The Game of Logic, has these two assumptions: 1\. "Logic is difficult or not many students like logic." 2\. "If mathematics is easy, then logic is not difficult." By translating these assumptions into statements involving propositional variables and logical connectives, de- termine whether each of the following are valid conclusions of these assumptions: a) That mathematics is not easy, if many students like logic. b) That not many students like logic, if mathematics is not easy. c) That mathematics is not easy or logic is difficult. d) That logic is not difficult or mathematics is not easy. e) That if not many students like logic, then either mathematics is not easy or logic is not difficult.

Use a proof by exhaustion to show that a tiling using dominoes of a \(4 \times 4\) checkerboard with opposite corners removed does not exist. [Hint: First show that you can assume that the squares in the upper left and lower right corners are removed. Number the squares of the original checkerboard from 1 to \(16,\) starting in the first row, moving right in this row, then starting in the leftmost square in the second row and moving right, and so on. Remove squares 1 and \(16 .\) To begin the proof, note that square 2 is covered either by a domino laid horizontally, which covers squares 2 and \(3,\) or vertically, which covers squares 2 and \(6 .\) Consider each of these cases separately, and work through all the subcases that arise. \(]\)

Suppose that the domain of the propositional function \(P(x)\) consists of the integers \(0,1,2,3,\) and \(4 .\) Write out each of these propositions using disjunctions, conjunctions, and negations. $$ \begin{array}{llll}{\text { a) }} & {\exists x P(x)} & {\text { b) } \forall x P(x)} & {\text { c) }} \quad {\exists x \neg P(x)} \\ {\text { d) }} & {\forall x \neg P(x)} & {\text { e) } \neg \exists x P(x)} & {\text { f) } \neg \forall x P(x)}\end{array} $$

Suppose that the domain of the propositional function \(P(x)\) consists of the integers \(1,2,3,4,\) and \(5 .\) Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions. $$ \begin{array}{ll}{\text { a) } \quad \exists x P(x)} & {\text { b) } \forall x P(x)} \\ {\text { c) } \quad \neg \exists x P(x)} & {\text { d) } \neg \forall x P(x)}\end{array} $$ e) \(\quad \forall x((x \neq 3) \rightarrow P(x)) \vee \exists x \neg P(x)\)
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