 # Proof of equivalence

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• #24363

Sir, this one is for the (ii)

To Prove:Congruence in an equivalence

Proof: Suppose we have an point A,A’ and A” in three triangles.

Since the triangles are congruent, the three points must be in the same place, making       congruence reflexive, transitive and symmetric.

#24365

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#24367

if a ~ a mod m then a-a is divided y m

if a ~ c mod m then a-c is divided y m so -(c-a) is divided y m

if a ~ c mod m , c ~ d mod m then a-c divided y m and c-d divided y m .. so a-c + c-d divided y m .. then a-d divided y m so a ~ d mod m

#24368

Proof of equivalence is :-

Equivalence relation has three critical properties –

1. Reflexive equivalence .eg of its use – A boy is related to himself (he is himself)
2. Symmetric equivalence .eg of his ues – Suman is the friend of mohit, then Mohit is the friend of Suman as well
3. Transitive equivalence. Eg of its use- Ram is the elder brother of Raju , Raju is the elder brother of Raj then Ram is the elder brother of Raj as well.

There are many methods of describing equivalence two of them are –

1. Gluing points is equivalence –
• How we can do this –  declare some points  and glue them together and then visualize in our mind what happens next
• Next we can check this by applying the three critical properties of equivalence-
• Reflexive – If we declare a point A then surely A is related to itself as it is itself A ~A
• Symmetric – If we declare two points A and B and glue them them together then A now becomes related to B as it becomes  B and B becomes A . A~B
• Transitive – If we declare three points A, B and C and glue them together then A becomes related to  B and C as it becomes B and C , B becomes A and C and C becomes A and B. A~B B~C thus A~C

Hence gluing points is an equivalence.

2. Congruence of numbers is an equivalence-

• We can prove this by lets say a number line we declare some points on the number line and check their distance by a  scale  of lets say 10 cm
• Next we can check this by applying the three critical properties of equivalence-
• Reflexive –  If we declare a point A then surely A to A’s distance is 0 cm which is divisible by 10 cm. A ~A
• Symmetric –  If we declare two points A and B on the number line and assuming that A B distance is multiples of 10 cm which is divisible by the scale . A ~B
• Transitive – If we declare three points A, B and C on the number line and assuming the distance between A and B ,B and C and A and C are multiples of 10 cm which is divisible by the scale. A~B B~C thus A~C

Thus  congruence of numbers is an equivalence theorem.

PROVED

#24369

Congruence of number lines:

Proof: Since we know that A is congruent to B, then B is congruent to A. (symmetric)

Since A is congruent to B, and B is congruent to C, then A is congruent to                                  C(transitive)

And A must be congruent to itself, being on the number line.(reflexive)

Thus the statement is proved 🙂

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