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Monaa123
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Given the set $\{1,2,3,4\},$ indicate whether it is reflexive, symmetric or transitive. $R = \{(1,1), (2,2), (3,3), (4,4)\}.$

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I said it's reflexive because $\forall x \in \{1,2,3,4\}$ there exists $(x,x) \in R.$

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I'm not sure if it's symmetric or transitive. The definition I was given of being symmetric is:

$\forall a,b \in X, aRb \Rightarrow bRa.$

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For transitive:

$\forall a,b,c \in X, aRb$ and $bRc \Rightarrow aRc.$

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I read the definition and I'm not sure if $a$ and $b$ can be the same number or not. I vaguely remember my lecturer saying a while ago that $a$ an $b$ can be the same number (I'm not sure what he was talking about when he said that though).\\

So if $a$ and $b$ can be the same number then it will be symmetric won't it?

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Could I use a similar argument for it being transitive?

So if a = 1, b = 1 and c = 1 then aRb and bRc $\Rightarrow$ aRc or can I not do that?
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poorform
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a and b can be the same number. For example think about the = relation on the reals. Clearly a~b if and only if a=b.

You are correct in saying it is reflexive as for any element in the set we have a~a.

Given the relation a~b then we know since the relation set is 1,1 2,2 3,3 4,4 that if a~b then a=b and it must be symmetric by the first property.

I believe you can apply the same argument for the proof of transitivity i.e if a~b and b~c then a=b=c and so a~c by the first property.
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Smaug123
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(Original post by Monaa123)


Given the set $\{1,2,3,4\},$ indicate whether it is reflexive, symmetric or transitive. $R = \{(1,1), (2,2), (3,3), (4,4)\}.$

\vspace{0.2in}



I said it's reflexive because $\forall x \in \{1,2,3,4\}$ there exists $(x,x) \in R.$

\vspace{0.2in}



I'm not sure if it's symmetric or transitive. The definition I was given of being symmetric is:

$\forall a,b \in X, aRb \Rightarrow bRa.$

\vspace{0.2in}



For transitive:

$\forall a,b,c \in X, aRb$ and $bRc \Rightarrow aRc.$

\vspace{0.2in}



I read the definition and I'm not sure if $a$ and $b$ can be the same number or not. I vaguely remember my lecturer saying a while ago that $a$ an $b$ can be the same number (I'm not sure what he was talking about when he said that though).\\

So if $a$ and $b$ can be the same number then it will be symmetric won't it?

\vspace{0.2in}



Could I use a similar argument for it being transitive?

So if a = 1, b = 1 and c = 1 then aRb and bRc $\Rightarrow$ aRc or can I not do that?
In fact, this has defined the finest equivalence relation on any set (in the sense that it has the most equivalence classes). x \sim y \iff x = y is an equivalence relation on any set, which is kind of the aim of this question. Intuitively it's clear, because we can quotient out by equivalence relations, and there's only one obvious thing we can quotient by to get the original set: namely, "identify every element with itself only".

ETA: "For all x,y in X" does mean "x may equal y". By convention, "for all distinct x,y" means "x may not equal y", but I prefer "for all x, y in X with x not equal to y", being the clearest.
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Monaa123
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(Original post by Smaug123)
In fact, this has defined the finest equivalence relation on any set (in the sense that it has the most equivalence classes). x \sim y \iff x = y is an equivalence relation on any set, which is kind of the aim of this question. Intuitively it's clear, because we can quotient out by equivalence relations, and there's only one obvious thing we can quotient by to get the original set: namely, "identify every element with itself only".

ETA: "For all x,y in X" does mean "x may equal y". By convention, "for all distinct x,y" means "x may not equal y", but I prefer "for all x, y in X with x not equal to y", being the clearest.
Aaah okay, I get you. Thanks!!
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