Date: Wed, 05 May 1999 11:12:41 -0500
Subject: equivalent relations

I am tutoring a boy who got this assignment from his teacher and I have no clue how to do it because I don't even know what the questions is asking! I need some help. Hereit is: "Give five examples of relations which are not equivalent relations and five examples of equivalent relations and explain why they are equivalent relations." This is seventh grade and I read about it in his book but it is not coming together for me. HELP!!!


Hi Megan

   A "relation" might be an intuitive notion, but it is rather hard to make precise what is meant by the word. (It is questionable whether kids should be forced to deal with such abstractions before they are ready for it. I was ready for it in my 4th year of university!)    To keep things simple, let us restrict our attention to number sets. Here are some examples of relations.

  1. x is related to y if "x and y have the same decimal value". So here 1/2 is related to 2/4. Also 2/4 is related to 1/2.

  2. x is related to y if "x is bigger than y". Here 5 is related to 3 but 3 is not related to 5. (This example shows that the technical notion of "related" in mathematics might ressemble our intuitive notion of a relationship, but they differ when it comes to details: among people, if a is related to b then necessarily b is related to a.)

  3. x and y are related if x^2 + y^2 = 1

  4. x and y are related if x + y < 1.
In general, we say that x and y are related if they satisfy a specific rule (called the relation). If they don't satisfy the rule then they are not related. The only requirement on relations is that the rule has to be clearly stated so that no matter which x and which y we are given, we are able to determine whether or not x and y are related. To generate lots of simple examples, just write down an equation or inequality using two variables. An "equivalence relation" is an important special case -- it allows us to make definitions. An equivalence relation arises when we decide that two objects are "essentially the same" under some criterion. A typical example from everyday life is color: we say two objects are equivalent if they have the same color. Thus a red fire truck and an apple would be equivalent using this criterion. Another typical example comes from the counting numbers: Any collection of 8 objects would be equivalent here -- thus a spider, an octopus, and a string quartet all have 8 legs and would be equivalent under this criterion.

   The technical definition:
An EQUIVALENCE RELATION is a relation on a set that satisfies three extra conditions:
   Every member of the set is related to itself.
   Whenever a is related to b it follows that b is related to a.
   Whenever a is related to b and b is related to c, it follows that a is related to c.

Examples using sets of numbers:

  1. Two fractions are equivalent if they equal the same decimal number. (or easier, a/b is equivalent to c/d if ad = bc.) (This is the only one of the examples in the list of relations given above that happens to be an equivalence relation.)

  2. Two counting numbers are equivalent if they have the same parity -- that is if they are both even or both odd. Here 4 is related to 6 (since both are even) but not to 7.

  3. Two counting numbers are equivalent if they leave the same remainder after division by 3. Here 4 is related to 7 (since both are 1 more than a multiple of 3) but not to 6.

  4. Throw numbers randomly into boxes. Say that two numbers are equivalent when they are in the same box.

  5. Equality is a very special equivalence relation. An number is equal only to itself and not to any other number. (It amounts to the box example #4 with exactly 1 number in every box.)


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