Hi,

I am taking a course in ring theory and i have problems with the following question:

a) How many different equivalence relations can be defined on the set X={a,b,c,d}?

i know a~a, b~b,c~c,d~d does that mean there are only 4 different equivalence relations?

b)Show that 6 divides the product of any 3 consecutive integers. I know it is true that 6 divides the product of any 3 consecutive integers. However, i have problem showing the proof.

Thank you and i do appreciate your help!

Eva

Hi Eva,

For your first question think in terms of partitions. An equivalence relation on a set generates a partition of the set and a partitiion of a set generates an equivalence relation. Thus you can ask "how many partitions are there of the set {a,b,c,d}?"

For a set with one element, {a}, there is one partitiion {{a}}.

For a set with two elements {a,b} there are two partitions {{a,b}} and {{a},{b}}.

For a set with three elements {a,b,c} we need to be more systematic.

• If there is a member of the partition with three elements then there is only 1 such partition, {{a,b,c}}.
  
  
• If the largest memeber of the partition has two elements then there are 3 such partitions, {{a,b},{c}}, {{a,c},{b}} and {{b,c},{a}}
  
  
• If the largerst element of the partition has only one element then there is only 1 such partition {{a},{b},{c}.

Now try a set with 4 elements.

With respect to your second question, consider that in order to be divisible by 6, an integer must be divisible by both 2 and 3. If you have 3 consecutive integers you have at least one that is even so the product must have a factor of 2 because of that integer. Also, for every 3 consecutive integers, one must be divisible by 3. A quick proof of this using modulus 3:

Thus the product n(n+1)(n+2) has a factor of 2 and a factor of 3 and so is always divisible by 6.

Hope this helps,

Leeanne and Penny

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