- #1

Buffu

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**(1):-**Reflexive :- A relation ##R : A \to A## is called reflexive if ##(a, a) \in R, \color{red}{\forall} a \in A##

**(2):-**Symmetric :- A relation ##R : A \to A## is called symmetric if ##(a_1, a_2) \in R \implies (a_2, a_1) \in R, \color{red}{\forall} a_1, a_2 \in A##

**(3):-**Transitive :- A relation ##R : A \to A## is called transitive if ##(a_1, a_2) \in R \land (a_2, a_3) \in R \implies (a_1, a_3) \in R, \color{red}{\forall} a_1, a_2,a_3 \in A##

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Here are examples in my textbook that i guess are violating 'for all' part of the definition.

##\large{\cdot}## Let ##R## be a relation in set ##\{ 1, 2, 3 \}##. ##R = \{(1,2), (2,1)\}##. It is given that this relation is symmetric.

My confusion :- Why is this symmetric ? It clearly violates the 'for all' part of the definition ##(2)##. It does not contain ##(1, 3), (3,1), (2,3), (3,2)##.

##\large{\cdot}## A relation ##R## on set ##\{ 1,2,3,4\}## given by ##R = \{(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2)\}## is both reflexive and transitive.

Same as one, Why is this thing transitive, it does not have ##(1,4), (4, 3), (1,3) \cdots ##.

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After a bit of thinking i reached the conclusion that definition for symmetric means that, If ##R## is symmetric then ##(a,b) \in R \implies (b,a)\in R## and it does not have to be that every doublet in ##A \times A## have to be in ##R##.

I reached the similar conclusions for reflexive and transitive definitions.

But then I saw this example :-

##R =\{(1,1)\}## is not reflexive for a set ##A = \{1,2\}##.And the reason given is that; ##R## is not reflexive because it does not contain ##(2,2)##.

So why can't i argue on the basis of the example that previous two examples of mine are not symmetric and transitive, respectively ?

All three definitions have ##\forall## in them but only for reflexive it really mean "for all" for, the rest it simply mean "for". Why is this so ?

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I think i am so confuse that some parts of this thread are not understandable by the readers.

So if the thread is too confusing to understand, I will glad if some one can explain why examples ##(1),(2)## are symmetric and transitive respectively when they clearly violates the "for all" part of the definition.