few false\true questions in set theory

[mathsnoob1]

Hello,

I've spent the last 5 hours solving a lot of false\true questions and i got several of which i am not sure. hoping you could help me with those:

(just tell me if you think the claim is true or false. it'd be very helpful if you can include a short explanation to help me understand and improve)

1)if R i a relation of A so

R^(-1)R=I_(A)

if and only if range(R) = A.

2)if R i a relation over set A and if

RR^(-1)=I_A

then range(R) = A.

3)for every relation R:

RR^(-1)=R^(-1)R

(really not sure about this one)

4)if R, S are relations over A so that SR = RS, so

S^(-1)R= RS^(-1)

5)if R is a reflexive relation over set A then

RR^(-1)=I_A

6)if R is anti symmetric then

R^(-1)

is antisymmetric (seems true and trivial, but i'm really not sure)

7)for every relation R, the relation

R \cup R^2

is transitive (seems true, is it?)

i've spent the last hours solving dozens of true\false and i'm not sure regarding those i've written. my due date is in a day and a half, so i'm hoping you could aid me with those.

if you can, in addition to writing if it's true or not, please give a little summary of the answer to help me understand better.

thank you very much for your help, i've been doing those for so many hours until i found this great place!

note: i don't know how to type math formulas, so i googled for mathjax tutorial and tried my best.


thanks again!

 

[ksdhart2]

Well, as you read in the Read Before Postinghttps://www.freemathhelp.com/forum/threads/41538-Read-Before-Posting!! thread, providing answers isn't what we do here. Being that you've shown no work of your own on these problems, I'll assume you have none to show. In light of that, here's a few leading questions to hopefully get you thinking in the right direction:

1) If R [is] a relation of A so \(\displaystyle R^{-1} \times R = I_A\) if and only if range(R) = A[/tex]

2) If R [is] a relation over set A and if \(\displaystyle R \times R^{-1} = I_A\) then range(R) = A.

These two exercises are basically the same, and if you get one, you'll pretty much automatically get the other. What is the definition of the cross product? What does it mean to take the Cartesian Product of a relation with its inverse? Being that the cartesian product, in general, isn't commutative, does changing the order of the arguments change the result here? Why or why not? What is the definition of the range of a relation? What does the notation \(\displaystyle I_A\) signify? What does it mean for a Cartesian Product to be equal to this?

3) For every relation R \(\displaystyle R \times R^{-1} = R^{-1} \times R\)

4) If R, S are relations over A so that SR = RS, so \(\displaystyle S^{-1} \times R = R \times S^{-1}\)

Again, these two exercises are very similar. They utilize the same leading question as the previous ones, re: the commutativity of the Cartesian Product. Maybe try picking a few sample relations and see where it takes you. Remember that to prove something is false, you need only one counterexample, but proving something is true needs to be more rigorous. Given the general non-commutativity of the Cartesian Product, what does it mean for the Cartesian Product to be commutative for two relations?

5) If R is a reflexive relation over set A then \(\displaystyle R \times R^{-1} = I_A\)

What does it mean for a relation to be reflexive? What does it mean to take the Cartesian Product of a relation with its inverse? How, if at all, does the relation being reflexive change the answer to this question?

6) If R is anti symmetric then \(\displaystyle R^{-1}\) is also anti-symmetric.

What does it mean for a relation to be anti-symmetric? Is is true that the inverse of any anti-symmetric relation is automatically also anti-symmetric? Again, recathat to prove something is false, you need only one counterexample, but proving something is true needs to be more rigorous.

7) For every relation R, the relation \(\displaystyle R \cup R^2\)

What does the notation \(\displaystyle R^2\) signify? What does it mean to take the union of a relation with this other relation? What does it mean for a relation to be transitive? Is the result of this union transitive?

Go ahead and give these exercises your best shot. If you get stuck again, that's okay, but when you reply back, please include any and all work you've done on these problems, even the parts you know for sure are wrong. Thank you.

 

[mathsnoob1]

sorry for not providing enough details. i'll tell you what i think and the explanations for it and please tell me if i'm right or wrong(it's very important for me to succeed and learn from my mistakes).

1) If R [is] a relation of A so [FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[/FONT] if and only if range(R) = A[/tex]

2) If R [is] a relation over set A and if [FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[/FONT] then range(R) = A.

since we're speaking here of a cartesian product, and since the result given is \(\displaystyle I_A\) i think that due to the definition of range(R) (image of function, the set of all the variables b in which (a,b) that belong to R), then 2) is right, but i don't know if range(R)=A is mandatory, though i think it's also correct. can you help me understand the difference and the right solution?

3) For every relation R [FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]R[/FONT]



4) If R, S are relations over A so that SR = RS, so [FONT=MathJax_Math]S[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]S[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT]

re

regarding 3) i think since as explained the cartesian product is not commulative, then it's not true at all(easier to show with simple example in relations).
regarding 4) i think that if the equation applies, it means that the product of the given relation,SR = RS might be \(\displaystyle I_A\), and then it would make sense. if i'm right, and SR = RS = \(\displaystyle I_A\) then the equation is true, else it's false.
again, please correct me if i'm wrong.

5) If R is a reflexive relation over set A then [FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[/FONT]fff


i think that the definition of reflexive relation is: ∀x ∈ X : x R x. so it seems to make sense and i believe it's true. x*1/x=1.

6If R is anti symmetric then [FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT] is also anti-symmetric.
since the definition of anti symmetric relation is: "if R(a,b) and R(b,a), then a = b", then i believe it is true, since the definition holds for R, and [FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT] is {(y,x) | xRy}, then i think it still applies.

7) For every relation R, the following relation is transitive: [FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]∪[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]2[/FONT]fffff is is

i am sorry that i mistakingly ommited the "transitive" part.
since we're speaking of union here, it includes elements of both. so based on the definition of transitivity(xRy and yRz implies xRz), i believe that it is true.

i supplied the reasoning and my best try to solve those. please help me improve by stating what is true and what is not, and if i did a mistake please explain it to me if you can. i learn less by riddles then by just correcting my mistakes.

thank you very much for helping.

[FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]R[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[/FONT] [FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[/FONT][/FONT][/FONT] [FONT=MathJax_Math][/FONT][FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]Affff[/FONT][FONT=MathJax_Math]I[/FONT][FONT=MathJax_Math]A[/FONT]