True or Flase? **Title: Understanding Relations and Functions with Arrow Diagrams****Introduction:**This exercise helps illustrate the concept of a relation $ R $ from set $ A $ to set $ B $, and explores when such a relation can also be considered a function. **Explanation of Sets:**- Let $ A = \{4, 5, 6\} $- Let $ B = \{6, 7\} $**Defining the Relation $ R $:**- The relation $ R $ from $ A $ to $ B $ is defined such that: $ (x, y) \in R $ means that $ x - y $ is an even integer.**Explanation of Even Integer Difference:**- For $ (x, y) \in R $, the difference $ x - y $ must be an even number. - An even integer is any integer that is divisible by 2 without a remainder.**Checking the Relation:**- For $ x = 4 $ - If $ y = 6 $, $ 4 - 6 = -2 $, which is even. - If $ y = 7 $, $ 4 - 7 = -3 $, which is not even.- For $ x = 5 $ - If $ y = 6 $, $ 5 - 6 = -1 $, which is not even. - If $ y = 7 $, $ 5 - 7 = -2 $, which is even.- For $ x = 6 $ - If $ y = 6 $, $ 6 - 6 = 0 $, which is even. - If $ y = 7 $, $ 6 - 7 = -1 $, which is not even.**Conclusion - Relation $ R $ as a Function:**- For $ x = 4 $, $ y $ must be 6.- For $ x = 5 $, $ y $ must be 7.- For $ x = 6 $, $ y $ must be 6. Thus, the relation $ R $ from $ A $ to $ B $ is indeed a function because each element in $ A $ is related to exactly one element in $ B $.** ### Proposition Logic and Formal StatementsLet $ L(x, y) $ be the proposition function "x loves y." Then the statement in informal language "Someone loves everybody" has its formal logical statement:\[ \exists x, \ \forall y, \text{ such that } L(x, y). \]This mathematical proposition can be translated into formal logic to clearly define the relationships and quantifiers involved in the statement. This logical form involves:- **Existential Quantifier ( $\exists$ )**: This symbol indicates that there exists at least one 'x' in the domain such that the following statement is true.- **Universal Quantifier ( $\forall$ )**: This symbol indicates that for all 'y' in the domain, the statement that follows holds true.Hence, the entire expression $ \exists x, \ \forall y, \text{ such that } L(x, y) $ can be read as "There exists a person 'x' such that for every person 'y,' the person 'x' loves the person 'y'."

Name: True or Flase? **Title: Understanding Relations and Functions with Arr
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Description: True or Flase? **Title: Understanding Relations and Functions with Arrow Diagrams****Introduction:**This exercise helps illustrate the concept of a relation $ R $ from set $ A $ to set $ B $, and explores when such a relation can also be consid

We have to perform the subtraction operation here on the 2 sets. If on subtraction every element of…

Math

Algebra

Let A = {4, 5, 6} and B = {6,7} and define a relation R from A to Bas: (x, y) E R means that x – y is an even integer. Then this relation R is actually a function from A to B. Hint: Draw an arrow diagram for this relation.

Let A = {4, 5, 6} and B = {6,7} and define a relation R from A to Bas: (x, y) E R means that x – y is an even integer. Then this relation R is actually a function from A to B. Hint: Draw an arrow diagram for this relation.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

True or Flase?

**Title: Understanding Relations and Functions with Arrow Diagrams**

**Introduction:**
This exercise helps illustrate the concept of a relation $ R $ from set $ A $ to set $ B $, and explores when such a relation can also be considered a function.

**Explanation of Sets:**
- Let $ A = \{4, 5, 6\} $
- Let $ B = \{6, 7\} $

**Defining the Relation $ R $:**
- The relation $ R $ from $ A $ to $ B $ is defined such that:
$ (x, y) \in R $ means that $ x - y $ is an even integer.

**Explanation of Even Integer Difference:**
- For $ (x, y) \in R $, the difference $ x - y $ must be an even number.
- An even integer is any integer that is divisible by 2 without a remainder.

**Checking the Relation:**
- For $ x = 4 $
- If $ y = 6 $, $ 4 - 6 = -2 $, which is even.
- If $ y = 7 $, $ 4 - 7 = -3 $, which is not even.
- For $ x = 5 $
- If $ y = 6 $, $ 5 - 6 = -1 $, which is not even.
- If $ y = 7 $, $ 5 - 7 = -2 $, which is even.
- For $ x = 6 $
- If $ y = 6 $, $ 6 - 6 = 0 $, which is even.
- If $ y = 7 $, $ 6 - 7 = -1 $, which is not even.

**Conclusion - Relation $ R $ as a Function:**
- For $ x = 4 $, $ y $ must be 6.
- For $ x = 5 $, $ y $ must be 7.
- For $ x = 6 $, $ y $ must be 6.

Thus, the relation $ R $ from $ A $ to $ B $ is indeed a function because each element in $ A $ is related to exactly one element in $ B $.

**

### Proposition Logic and Formal Statements

Let $ L(x, y) $ be the proposition function "x loves y." Then the statement in informal language "Someone loves everybody" has its formal logical statement:

\[ \exists x, \ \forall y, \text{ such that } L(x, y). \]

This mathematical proposition can be translated into formal logic to clearly define the relationships and quantifiers involved in the statement. This logical form involves:

- **Existential Quantifier ( $\exists$ )**: This symbol indicates that there exists at least one 'x' in the domain such that the following statement is true.
- **Universal Quantifier ( $\forall$ )**: This symbol indicates that for all 'y' in the domain, the statement that follows holds true.

Hence, the entire expression $ \exists x, \ \forall y, \text{ such that } L(x, y) $ can be read as "There exists a person 'x' such that for every person 'y,' the person 'x' loves the person 'y'."

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