Proving Equivalency Relations: Help from Henry

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Therefore, (x + y) S (x' + y').In summary, the relation S on the set of real numbers is an equivalence relation, as it is reflexive, symmetric, and transitive. Furthermore, if x S x' and y S y', then (x + y) S (x' + y'), making S a congruence with respect to addition.
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henry1
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I'm having copious amounts of trouble with this question and an amount of help would really be appreciated.

Let S be the relation on the set of real numbers defined by

x S y iff x-y is an integer

1. prove that S is an equivalence relation on R.

2. Prove that if x S x' and y S y' then (x+y) S (x'+y').

Thanks,

Henry.
 
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Can you show what you have tried so far so our helpers can see where you are stuck or may be going astray?
 
  • #3
My thoughts:

S is reflexive because 0 is an integer.

S is symmetric because -k is an integer whenever k is.

S is transitive, because the sum of two integers is another integer.

The second part of the problem is to show S is a congruence with respect to addition (of real numbers). This really just amounts to working through the definition of S:

Suppose x S x'. Then x - x' = k, for some integer k. Similarly, y S y' means y - y' = m, for some integer m.

Consequently:

(x + y) - (x' + y') = (x - x') + (y - y') = k + m, which is, of course, an integer.
 

FAQ: Proving Equivalency Relations: Help from Henry

What is an equivalency relation?

An equivalency relation is a relation on a set that is reflexive, symmetric, and transitive. This means that every element in the set is related to itself, if any two elements are related then they are symmetrically related, and if two elements are related and one is related to a third element, then the first and third element are also related.

How do you prove that a relation is an equivalency relation?

In order to prove that a relation is an equivalency relation, you must show that it satisfies the three properties of reflexivity, symmetry, and transitivity. This can be done through mathematical proofs or by providing specific examples that demonstrate the properties.

What is the role of Henry in proving equivalency relations?

Henry is a fictional character that is used as an example to help understand the concept of equivalency relations. His actions and relationships with other characters can be used to demonstrate the properties of an equivalency relation.

Can two different equivalency relations exist on the same set?

Yes, it is possible for two different equivalency relations to exist on the same set. For example, the relation "is the same age as" and the relation "is the same height as" can both be equivalency relations on the set of people.

How are equivalency relations used in mathematics and science?

Equivalency relations are used in mathematics and science to analyze and understand relationships between different elements. They can be used to prove theorems, classify objects, and identify patterns. In science, equivalency relations can help us understand the properties and behavior of different substances and systems.

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