/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 21\. For the following statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 21

21\. For the following statements the universe comprises all nonzero integers. Determine the truth value of each statement. a) \(\exists x \exists y[x y=1]\) b) \(\exists x \forall y[x y=1]\) c) \(\forall x \exists y[x y=1]\) d) \(\exists x \exists y[(2 x+y=5) \wedge(x-3 y=-8)]\) e) \(\exists x \exists y[(3 x-y=7) \wedge(2 x+4 y=3)]\)

Short Answer

Expert verified
The truth values of the given logical statements are: a) True b) False c) True d) True e) False

Step by step solution

01

Evaluate Statement a

Statement a says, 'there exist numbers x and y such that their product is 1'. This is true. For instance, if x and y are both 1, their product is indeed 1.
02

Evaluate Statement b

Statement b says, 'there exists a number x, for which all numbers y when multiplied with x give 1'. This is false. The only number which when multiplied with all other numbers gives 1 is 1 itself. But for integers other than 1, the condition fails, hence the statement is false.
03

Evaluate Statement c

Statement c states 'for every x, there exists a y such that their product is 1'. This is true because for every non-zero integer x, its reciprocal (1/x) which is the y in this case, gives the product 1.
04

Evaluate Statement d

Statement d is a system of linear equations with 2 variables x and y. Using substitution or elimination method, we find that there is a solution for x and y which satisfies both equations. Therefore, this statement is true.
05

Evaluate Statement e

Statement e is also a system of linear equations with 2 variables x and y. Solving the equations, we find that there is no pair of x, y that can satisfy both equations at the same time. Therefore, this statement is false.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Quantifiers in Discrete Mathematics
Quantifiers are symbols used in logic and mathematics to express the extent to which a statement is true, often dealing with sets or large quantities of elements. They help us specify constraints or guidelines for statements involving a range of items. In logic, the two main types of quantifiers are existential and universal.

  • **Existential Quantifier (\(\exists\))**: Indicates there is at least one element in the universe that satisfies a particular condition. For example, \(\exists x\) means 'there exists an \(x\)' such that a particular proposition is true.
  • **Universal Quantifier (\(\forall\))**: Suggests that a specified property holds for all elements in the considered set. For example, \(\forall x\) implies 'for all \(x\)' in a given set, a specified condition holds true.
When evaluating expressions with quantifiers, one must consider the domain of discourse, which contains all possible values that variables can assume. In the exercise provided, the universe is all nonzero integers. This means that any element considered under these quantifiers must be a nonzero integer.

By understanding and applying these quantifiers effectively, we can derive precise logical conclusions from statements, tackling various problems from simple assertions to complex mathematical proofs.
Systems of Linear Equations in Discrete Math
A system of linear equations involves two or more linear equations containing multiple variables. The ultimate goal is to find values for these variables that satisfy all equations simultaneously.

The diagrammatic representation is often a set of lines in a multi-dimensional space (like the x-y coordinate plane for 2 variables). Each equation represents one line, and solutions are the points where these lines intersect.

To solve such systems, we use methods like:
  • **Substitution**: Solve one equation for one variable and substitute that into another equation.
  • **Elimination**: Add or subtract equations to eliminate a variable, making it easier to solve.
The solution to a system can fall into three categories:
  • **One Solution**: Lines intersect at a single point.
  • **No Solution**: Lines are parallel and never intersect.
  • **Infinitely Many Solutions**: Lines coincide, meaning they lie on top of each other fully overlapping.
Understanding these systems' solutions is crucial for applications in discrete math, as they provide insights into solving real-world problems that utilize constraints and relationships among variables.
Explaining Truth Values in Logic
Truth values are fundamental components in logic, where they assess whether statements are true or false. This binary approach enables clear decision-making in logical operations. The truth value of a statement is crucial for evaluating logical expressions, especially when quantifiers are involved.

Logical expressions determine truth values based on the validity of their components. For example, if a statement using a universal quantifier, like \( \forall x [P(x)] \), is true, it means every possible instance of \( x \) makes \( P(x) \) true. If even one instance makes \( P(x) \) false, the whole statement is false.

For existential quantifiers, such as \( \exists x [P(x)] \), if at least one value of \( x \) makes \( P(x) \) true, then the statement holds true. Otherwise, it is false. In the context of our exercise, determining the truth value involves checking if specific conditions exist within the given universe.

Ultimately, grasping truth values empowers us to interpret logical arguments accurately and solve mathematical problems by confirming the validity of proposed statements.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

19\. For each of the following statements state the converse, inverse, and contrapositive. Also determine the truth value for each given statement, as well as the truth values for its converse, inverse, and contrapositive. (Here "divides" means "exactly divides.") a) [The universe comprises all positive integers.] If \(m>n\), then \(m^{2}>n^{2}\) b) [The universe comprises all integers.] If \(a>b\), then \(a^{2}>b^{2}\). c) [The universe comprises all integers.] If \(m\) divides \(n\) and \(n\) divides \(p\), then \(m\) divides \(p\). d) [The universe consists of all real numbers.] \(\forall x\left[(x>3) \rightarrow\left(x^{2}>9\right)\right]\) e) [The universe consists of all real numbers.] For all real numbers \(x\), if \(x^{2}+4 x-21>0\), then \(x>3\) or \(x<-7\)

9\. Write the converse, inverse, and contrapositive of each of the following implications. For each implication, determine its truth value as well as the truth values of its corresponding converse, inverse, and contrapositive. a) If \(0+0=0\), then \(1+1=1\). b) If \(-1<3\) and \(3+7=10\), then \(\sin \left(\frac{3 \pi}{2}\right)=-1\).

7\. For the universe of all integers, let \(p(x), q(x), r(x), s(x)\), nd \(t(x)\) be the following open statements. \(s(x) ; \quad x\) is (exactly) divisible by 4 \(t(x): \quad x\) is (exactly) divisible by 5 a) Write the following statements in symbolic form. i) At least one integer is even. ii) There exists a positive integer that is even. iii) If \(x\) is even, then \(x\) is not divisible by 5 . iv) No even integer is divisible by 5 . v) There exists an even integer divisible by \(5 .\) vi) If \(x\) is even and \(x\) is a perfect square, then \(x\) is divisible by 4 b) Determine whether each of the six statements in part (a) is true or falsc. For cach false statement, provide a counterexample. c) Express each of the following symbolic representations in words. i) \(\forall x[r(x) \rightarrow p(x)]\) ii) \(\forall x[s(x) \rightarrow q(x)]\) iii) \(\forall x[s(x) \rightarrow \neg t(x)]\) iv) \(\exists x[s(x) \wedge \neg r(x)]\) d) Provide a counterexample for each false statement in part (c)

15\. Define the connective "Nand" or "Not ... and ..." by \((p \uparrow q) \Leftrightarrow \neg(p \wedge q)\), for any statements \(p, q .\) Represent the following using only this connective. a) \(\neg p\) b) \(p \vee q\) c) \(p \wedge q\) d) \(p \rightarrow q\) e) \(p \leftrightarrow q\)

4\. Let \(p, q, r, s\) denote the following statements: \(p\) : I finish writing my computer program before lunch. q: I shall play tennis in the afternoon. \(r\) : The sun is shining. s: The humidity is low. Write the following in symbolic form. a) If the sun is shining. I shall play tennis this afternoon. b) Finishing the writing of my computer program before lunch is necessary for my playing tennis this afternoon. c) Low humidity and sunshine are sufficient for me to play tennis this afternoon.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.