Thread: Clarifying what this is asking (Reflexive, Symmetric, AntiSymmetric, and Transitive)

So I get the basics of these four topics (reflexive, symmetric, antisymmetric, and transitive). I understand the rules in which they work but for some I am having a hard time figuring out what exactly this question is asking.

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y) is an element of R if and only if:

A) x + y = 0
B) x = + or - y
C) x - y is a rational number.
D) xy = 0
...

And so forth.

So is the question asking if the elements of R are only elements if (for the case of D) ) they satisfy the condition of xy = 0. So if that is the case, (1,1) would not be in my set and is not taken into consideration in answering the 4 questions?

Okay so re reading what I just typed can be confusing so I am going to post one of my answers.

For A) I have:

Reflexive: No: Counter example: (1,1) is an element of R but 1 + 1 does not equal 0. Therefore, it cannot be reflexive.

Now I am not sure if (1,1) can actually be taken into consideration because to satisfy the condition of x + y = 0, the point (1,1) would not be used.

I have a feeling I am over thinking the problem but this is frustrating me beyond control. Thanks for the response!

Originally Posted by YaegerBomb

So I get the basics of these four topics (reflexive, symmetric, antisymmetric, and transitive). I understand the rules in which they work but for some I am having a hard time figuring out what exactly this question is asking.

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y) is an element of R if and only if:

A) x + y = 0
B) x = + or - y
C) x - y is a rational number.
D) xy = 0
...

And so forth.

So is the question asking if the elements of R are only elements if (for the case of D) ) they satisfy the condition of xy = 0. So if that is the case, (1,1) would not be in my set and is not taken into consideration in answering the 4 questions?

Okay so re reading what I just typed can be confusing so I am going to post one of my answers.

For A) I have:

Reflexive: No: Counter example: (1,1) is an element of R but 1 + 1 does not equal 0. Therefore, it cannot be reflexive.

Now I am not sure if (1,1) can actually be taken into consideration because to satisfy the condition of x + y = 0, the point (1,1) would not be used.

I have a feeling I am over thinking the problem but this is frustrating me beyond control. Thanks for the response!

no, for A, (1,1) is not in the relation, so you can't use that.

just go through and check the definitions.

For (A)

we have x + y = 0

=> x = -y

(i) reflexive means (x,x) is in R. does x = -x? No, not in general. this only happens if x = 0, and so it does not work for all real numbers. the relation is not reflexive.


(ii) symmetric means (x,y) is in R implies that (y,x) is in R.

Assume (x,y) in R, then x = -y. Multiplying both sides by -1 we get y = -x, that is, (y,x) is in R. The relation is symmetric.


(iii) anti-symmetric means if (x,y) and (y,x) is in R, then x = y.

The relation is not anti-symmetric. Counter-example, (1,-1) is in R. Clearly x = -y and y = -x but x and y are distinct.


(iv) transitive means if (x,y) is in R and (y,z) is in R, then (x,z) is in R.

The relation is not transitive. Counter-example: (1,-1) and (-1,1) is in R, but (1,1) is not.


Now do the others the same way. let the definitions guide you

Originally Posted by Jhevon

no, for A, (1,1) is not in the relation, so you can't use that.

just go through and check the definitions.

For (A)

we have x + y = 0

=> x = -y

(i) reflexive means (x,x) is in R. does x = -x? No, not in general. this only happens if x = 0, and so it does not work for all real numbers. the relation is not reflexive.


(ii) symmetric means (x,y) is in R implies that (y,x) is in R.

Assume (x,y) in R, then x = -y. Multiplying both sides by -1 we get y = -x, that is, (y,x) is in R. The relation is symmetric.


(iii) anti-symmetric means if (x,y) and (y,x) is in R, then x = y.

The relation is not anti-symmetric. Counter-example, (1,-1) is in R. Clearly x = -y and y = -x but x and y are distinct.


(iv) transitive means if (x,y) is in R and (y,z) is in R, then (x,z) is in R.

The relation is not transitive. Counter-example: (1,-1) and (-1,1) is in R, but (1,1) is not.


Now do the others the same way. let the definitions guide you

Wow thanks for clarifying that for me. I used this same method for the next thousand problems (or at least it felt like it) I had to do.