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English translation needed :)

**Functions and Relations: Symmetric, Transitive, Reflexive.**

Hello! I need an English translation for these questions. I've come across YouTube videos and they all have really simple examples, but then again my homework gets complicated for me to understand. :)

Once I get the symbols understood then I might be able to find out the others. Thanks so much!

Attachment 28847

Attachment 28848

Re: English translation needed :)

this is all i could come up with

Curly R = Relation

C = equal subset

| = divides

Z+ = any positive integer(?)

Z = ?

Curly U = universe

Curly P =

C = ?

R^2 = ?

Re: English translation needed :)

This is all I could translate by my own understanding. But actually it doesn't make sense to me; so I can't really find out whether their symmetric, reflexive, transitive... or whatever. Please correct me! :)

a. Relation R is an equal subset of Z+ x Z+ (two sets of positive integers; paired) where a relation of "a and b" if a divides b.

b. Relation R is the relation on set Z where a relation of "a and b" if a divides b.

c. For a given universe U and a fixed subset of C of U, define the relation R and Powerset of U as follows: For A, B is an equal subset of the universe we have a Relation of A and B if .....?

d. On te set of A of all lines in R^2, define the relation R for two lines l1, l2 by the relation R of "l1 and l2" if l1 is perpendicular to l2.

e. Relation R is the relation of set Z where the Relation of "x and y" if x + y is odd.

f. Relation R is the relation on set Z where the relation of "x and y" if x-y is even.

g. Let set T be the set of all triangles in R^2. Define Relation R on set T by the relation of (t1 and t2) have an angle of the same measure.

h. Relation R is the relation on Z x Z (set Z is paired to itself?) where the relation of (a,b) and (c,d) if a is less than or equal to c.

Re: English translation needed :)

I'm trying to solve A. I am guessing that I'll make up sets with numbers in them. Here's my try.

Given

R __c__ Z+ x Z+

(a,b)

a|b

let a = {2}

let b = {2, 4, 6, 8, 10}

Z+ x Z+ would come out as:

{ (2 , 2), (2 , 4), (2 , 6) , (2 , 8) , (2 , 10) , (4 , 2), (6,2) , (8, 2) , (10 , 2) }

Symmetric:

{ (2 , 2), (2 , 4), (2 , 6) , (2 , 8) , (2 , 10) , (4 , 2), (6,2) , (8, 2) , (10 , 2) }

Reflexive:

{ (2 , 2)}

Transitive?

I'm not even sure if I did the right thing. :(

Re: English translation needed :)

Quote:

Originally Posted by

**jpab29** this is all i could come up with

Curly R = Relation

C = equal subset

| = divides

Z+ = any positive integer(?)

Z = ?

Curly U = universe

Curly P =

C = ?

R^2 = ?

You are really confused as to notation.

Consider a) $\displaystyle \forall k\in\mathbb{Z}^+[~k|k]$. Every positive integer divides itself: reflective.

Consider b) $\displaystyle 0\not |~ 0$; Zero is not a divisor of anything: not reflexive.

Consider d) No line is perpendicular to itself. What does that tell you?

If $\displaystyle \ell_1\ne\ell_2$ then $\displaystyle {\ell _1} \bot {\ell _2}\; \Leftrightarrow \;{\ell _2} \bot {\ell _1}$ . What does that tell you?

Re: English translation needed :)

Thank you so much Plato!

Yes I am very confused with notation.

is there a way that you can make a) symmetric, antisymmetric or transitive? or is just reflective alone?

and for these particular questions, do I make up numbers to show that it's symmetric, transitive etc? Because if it us letters alone, I can't tell the difference. :(

As to no line is perpendicular to itself, L1 cannot be perpendicular to L1. So L1 has to be paired with another line L2, to make it possibly perpendicular. So it is not reflexive, but it is symmetric.

Re: English translation needed :)

Quote:

Originally Posted by

**jpab29** Thank you so much Plato!

Yes I am very confused with notation.

is there a way that you can make a) symmetric, antisymmetric or transitive? or is just reflective alone?

You know that $\displaystyle 2\,|\;\4$ BUT $\displaystyle 4\not |\;2$ so it cannot be symmetric.

Now $\displaystyle k\;|\;j\text{ and }j\;|\;k\text{ if and only if }k=j$, so it is antisymmetric.

You show us that it is transitive.

Re: English translation needed :)

True.

2|4 where 2 divides 4 is possible, which the answer is 2.

4|2 where 4 divides 2 results into a fraction or decimal point number, which is not right.

So they are not symmetric.

To make it symmetric, do i have the freedom choose different numbers?

3|3, vice versa, will always output the same answer, will this be symmetric? More like reflexive....

To make it transitive...

I know by definition transitive is x is related to y, y is related to z, therefor x is related to z.

But i have no idea how to make it a|b to be transitive. :( There has to be 3 pairs of a|b; and I only learned how to do 2 pairs only. :(

Re: English translation needed :)

I don't know what you mean by "make it symmetric". In any case, you are not asked to do that, only to determine whether it is symmetric or not.

As for "transitive", that is "if aRb and bRc then aRc". Here, that would be "if a divides b and b divides c, then a divides c.

If a divides b then b= ai for some integer i. If b divides c, then c= bj for some integer j. Combining those, c= (ai)j= a(ij).

Re: English translation needed :)

Quote:

Originally Posted by

**HallsofIvy** I don't know what you mean by "make it symmetric". In any case, you are not asked to do that, only to determine whether it is symmetric or not.

As for "transitive", that is "if aRb and bRc then aRc". Here, that would be "if a divides b and b divides c, then a divides c.

If a divides b then b= ai for some integer i. If b divides c, then c= bj for some integer j. Combining those, c= (ai)j= a(ij).

That is where I get confused. When it is symmetric, reflexive or transitive. :( It is because i do not understand the notations of the statements fully. So sorry. It has been on my mind to make up numbers as sets to representa the letters that is stated in the statements. Just like how we made up 2|4 or 4|2. If i involve numbers, i will be less confused, because if it's just all letters (such as the statements itself) i get confused because I can't put any value in them.

As for a|b, b|c, therefor a|c. Can we plug in numbers for those too? I was unable to do it because i was already thinking of putting values on those letters. Because a,b and c can be any number right? So depending on the numbers we'll use, it will depend if it is symmetric, reflexive or transitive. And to make it transitive, i could not think of the correct numbers to plug-in in a for it to divide b, and b to divide c, so that a can divide c.

Re: English translation needed :)

Quote:

Originally Posted by

**jpab29** As for a|b, b|c, therefor a|c. Can we plug in numbers for those too? I was unable to do it because i was already thinking of putting values on those letters. Because a,b and c can be any number right? So depending on the numbers we'll use, it will depend if it is symmetric, reflexive or transitive. And to make it transitive, i could not think of the correct numbers to plug-in in a for it to divide b, and b to divide c, so that a can divide c.

If you are of a totally literal mind to the point of having use concrete examples (number, particular sets, people or whatever) then you never understand relations. Now that 'never' mat sound like a hard saying but it is true. It matter of common body of knowledge that if a divides b and b divides c then a divides c. No special knowledge is needed for that fact to be understood. "taller than", "subset of", "to the of in a line" are all **transitive relations**. No special knowledge or taking examples are needed for those facts.

Re: English translation needed :)

Quote:

Originally Posted by

**Plato** If you are of a totally literal mind to the point of having use concrete examples (number, particular sets, people or whatever) then you never understand relations. Now that 'never' mat sound like a hard saying but it is true. It matter of common body of knowledge that if a divides b and b divides c then a divides c. No special knowledge is needed for that fact to be understood. "taller than", "subset of", "to the of in a line" are all **transitive relations**. No special knowledge or taking examples are needed for those facts.

Yes that's true. I understand better with concrete examples. It's what my textbook has been doing in its examples.

It states the statement, then uses random numbers and it gets me all confused where such numbers come from. That's where i got the idea if i get simpler concrete examples, I'd understand better.

By the way, i'm on an online math class. I literally have no instructor to break it down for me, i just have my almost-impossible-to-understand textbook, YouTube and math forums hahAha

But last night i sent my instructor pages and pages of questions. I hope he will answer them appropriately, or else like what u said , I'll never understand relations.

Re: English translation needed :)

Quote:

Originally Posted by

**jpab29** By the way, i'm on an online math class. I literally have no instructor to break it down for me, i just have my almost-impossible-to-understand textbook,

All the studies show that on-line mathematics courses are useless unless the student has done the equivalent of *advanced placement* (AP) courses.

Re: English translation needed :)

Quote:

Originally Posted by

**Plato** All the studies show that on-line mathematics courses are useless unless the student has done the equivalent of *advanced placement* (AP) courses.

hahahaha! Right on! I'm with you on that. As to why I am on online class, it's because there are no universities in my area, except online. I'm married and pregnant so I can't travel often for on-campus classes which is hours away from where I am. Oh well, thanks for the help everyone! I will get some sleep and work on the other problems later today. I didn't sleep at all!

I appreciate all your help!

Re: English translation needed :)

Quote:

Originally Posted by

**Plato** All the studies show that on-line mathematics courses are useless unless the student has done the equivalent of *advanced placement* (AP) courses.

Here is an update to my post about online courses.