CMI2015-A-02

Question

A binary relation $R ⊆ (S ×S)$ is said to be Euclidean if for every $a, b, c ∈ S, (a, b) ∈ R$ and $(a, c) ∈ R$ implies $(b, c) ∈ R$. Which of the following statements is valid?

• A. If $R$ is Euclidean, $(b, a) ∈ R$ and $(c, a) ∈ R$, then $(b, c) ∈ R$, for every $a, b, c ∈ S$
• B. If $R$ is reflexive and Euclidean, $(a, b) ∈ R$ implies $(b, a) ∈ R$, for every $a, b ∈ S$
• C. If $R$ is Euclidean, $(a, b) ∈ R$ and $(b, c) ∈ R$, then $(a, c) ∈ R$, for every $a, b, c ∈ S$
• D. None of the above.

Comments

what is Euclidian?

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Why option C is not true?

Isn't it the same statement in the question?

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No they are not the same.

Answer 1

Lets see option $B$

If $R$ is reflexive then $(a,a)$ belongs to $R$

It is given that $(a,b)$ belongs to $R$

So, $(a,b) ∈ R$ and $(a,a)$ ∈ $R$ that implies $(b,a) ∈ R$ for all $a,b ∈ S$ [ since $R$ is euclidean]

Thus option $B$ is the correct answer.

Comments

Why you have checked only option B, what about option A and C?

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It is a special property, which called as Euclidean. C) is telling about transitivity property. But transitivity property is not Euclidean

Similarly A) also not a valid Euclidean

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Yeah, got it! Thank you

Answer 2