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Proof of equivalence

Home Forums Math Olympiad, I.S.I., C.M.I. Entrance Math Circle Proof of equivalence

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• #24350
Ashani Dasgupta
Keymaster

Show rigorously

1. Gluing points is equivalence

2. Congruence is an equivalence.

#24358

1) Gluing point is Equivalance

• Any point can be glued to same point — Reflexive (A~A)
• If point A is Glued to point B => Point B is glued to Point A — Symmetric (A~ B)
• If Point A is Glued to Point , Point B is glued to Point C, means Point A is glued to C — Transitive (A~ B, B~ C then A~ C).

Hence Gluing point on a plane is Equivalance.

2) Congruance is Equivalance

• Triangle A is congruent to Triangle A — Reflexive (A ~ A).
• If Triangle A is congruent to Triangle B => Triangle B is congruent to Triangle A — Symmetric (A ~ B)
• If Triangle A is congruent to Triangle B , Triangle B is congruent to Triangle C, means Triangle A is congruent to C — Transitive (A~ B, B~ C then A~ C).

Hence Congruance is Equivalance

#24360
Writaban Sarkar
Participant

1. We first check if gluing points is reflexive. If we have a point A , then A glued to A itself, since it is the point itself. Now we check if gluing points is symetric. If we have A is glued to B then we should have that B is glued to A. Now atlast we check if it is transitive. If A is glued to B and B is glued to C, then we have  A is glued to C, since both are glued to B. So we find that gluing points is an example of equivalence, since it is true for all 3 cases.

2. We see that if we have a triangle ABC then it ABC is congruent, since it approves SSS Congruence. Now if we have 2 triangles ABC and DEF  such that ABC is congruent is DEF then we also see that DEF is congruent is ABC. Now if we have three triangles ABC, DEF and IJK such that ABC is congruent DEF and DEF is congruent IJK, then we have ABC is congruent IJK, since they are both congruent to DEF.  So we find that congruence is an example of equivalence, since it is true for all 3 cases.

Therefore, we see that both gluing and congruence are examples of equivalence, since they are true for all 3 cases.

#24361
Gouri Basak
Participant

Now we have to prove that :

1- Gluing points is equivalence

2-Congruence is equivalence

Now to just prove that some fact is equivalence,we just need to check it if it is reflexive,symmetric and transitive.

Proof 1 – Gluing points is equivalence

We have to see that gluing points are reflexive,symmetric and transitive or not.(If they aren’t then it already tells us that gluing points is not equivalence)

i) It is reflexive as any point A glued to itself is equal

ii) It is symmetric as if any point A is glued to B it means the same when Point B is glued to A

iii) It is also transitive as if any point A is glued to B and that is glued to C it also means the same that Point C is glued to Point A

So it proves that gluing points is equivalence.

Proof 2- Congruence is equivalence

Like in the previous proof,we just have to check if it is reflexive,symmetric and transitive

i) It is reflexive as any object is congruent to itself

ii) It is symmetric as when an object A is congruent to object B then it means the same when the object B is congruent to object A

iii) It is transitive as when an object A is congruent to object B and that is congruent to object C then it also means the same that object C is congruent to object A

So it also proves that congruence is an equivalence.

#24362
Shivani Ramesh
Participant
• A point is equivalent only when it has 3 critical properties:
• symmetric(a~a)
reflexive(a~b and b~a)
transitive(a~b , b~c , c~a)
• We tell some points as same when we glue those points.
• When we glue these points in a certain manner we get a 3 dimensional figure.
• Congurence is equivalence
• It is reflexive as anything is congurent to itself.
• Suppose any number is congurent to another number it will be congurent to it.
• Since a~b,b~c,a~c.
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