# Equivalence Relations

• Jun 14th 2008, 08:32 AM
robocop_911
Equivalence Relations
I am not sure how to go about doing the following questions... any help will be highly appreciated..

Q. Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack.
a) {(a,b)/ a and b are the same age}
b) {(a,b)/ a and b have the same parent}
c) {(a,b)/ a and b share a common parent}
d) {(a,b)/ a and b have met}
e) {(a,b)/ a and b speak a common language}

a) a and b are same age, therefore it is reflexive since a = b
It is symmetric a = b and b = a, it is transitive since a = b and b = c implies a = c.

b) a and b have same parent, therefore it is reflexive since a = b, symmetric since a = b and b = a, but NOT transitive since a = b and b = c, but a != c.

Similarly the rest of them... but I don't have a clue how to go about doing it...
• Jun 14th 2008, 08:37 AM
Plato
Be careful! (b) & (c) look alike but are not.
(b) is transtive but (c) is not. WHY?

As a rule of thumb: the word "same" implies an equivlance relation.
• Jun 14th 2008, 08:47 AM
robocop_911
Quote:

Originally Posted by Plato
Be careful! (b) & (c) look alike but are not.
(b) is transtive but (c) is not. WHY?

As a rule of thumb: the word "same" implies an equivlance relation.

I was wondering if you could possibly explain me step-by-step.
How to attack these problems? Where to start?
• Jun 14th 2008, 09:05 AM
Plato
Let’s change (b) to “have the same mother”.
It is easy to see reflexive and symmetric.
If a & b have the same mother and b & c have the same mother then surly a & c have the same mother. So it is transitive.

But have a parent in common? Here a & b may have the same mother and b & c have the same father, but a & c may not have a common parent. Part (e) is exactly the same idea because some people are bilingual. A may speak English, B may speak English and French and C may speak French.
• Jun 14th 2008, 09:15 AM
robocop_911
Quote:

Originally Posted by Plato
Let’s change (b) to “have the same mother”.
It is easy to see reflexive and symmetric.
If a & b have the same mother and b & c have the same mother then surly a & c have the same mother. So it is transitive.

But have a parent in common? Here a & b may have the same mother and b & c have the same father, but a & c may not have a common parent. Part (e) is exactly the same idea because some people are bilingual. A may speak English, B may speak English and French and C may speak French.

It's easy for you to see if it is reflexive and symmetric but not for me!

Thanks for you help!
• Jun 14th 2008, 09:34 AM
Plato
Quote:

Originally Posted by robocop_911
It's easy for you to see if it is reflexive and symmetric but not for me!

Does a have the same mother as a, for every a? Reflexive.
If a & b have the same mother the b & a have the same mother? Symmetric.
It is easy.
• Jun 14th 2008, 09:42 AM
robocop_911
Quote:

Originally Posted by Plato
Does a have the same mother as a, for every a? Reflexive.
If a & b have the same mother the b & a have the same mother? Symmetric.
It is easy.

What about the rest of the questions? They are not the same as the above one.
• Jun 14th 2008, 09:58 AM
Plato
Quote:

Originally Posted by robocop_911
What about the rest of the questions? They are not the same as the above one.