What is the Union and Intersection of Modulo Relations?

[nistaria]

Homework Statement

Let R₁ and R₂ be the "congruent modulo 3" and "congruent modulo 4" relations on the set of integers.

Homework Equations

Find:
a) R₁ $$\cup$$R₂
b)R₁ $$\cap$$ R₂
There is also problem c, d but I won't write these here. If I am able to solve this, then the rest should be cake.

The Attempt at a Solution

my problem with this question is this: I'm not sure I understand what the question wants
Does it want all possible elements in the relation?
such as for a)
R₁ $$\cup$$R₂ ={(a,b)| (3|a-b) or (4|a-b)}
Is that a valid answer?
b) R₁ $$\cap$$ R₂= {(a,b)| 12|a-b}

PS: I have already proved in a previous problem that
a$$\equiv$$b(mod m) is an equivalence relation as it is transitive, symmetric and reflexive.

Thanks for reading

 

[tiny-tim]

welcome to pf!

hi nistaria! welcome to pf!

yes, that all looks fine

(a relation on X is just a subset of X x X, so yes you use the ordinary set union and intersection)

… the rest should be cake.

hmm … candy, coffee, and now cake …

are you food-motivated? ​

 

[nistaria]

hi nistaria! welcome to pf!

yes, that all looks fine

(a relation on X is just a subset of X x X, so yes you use the ordinary set union and intersection)


hmm … candy, coffee, and now cake …

are you food-motivated? ​

AHAHAHAHA
I guess you read my other posts! Give me food and you got yourself a happy camper!

According to this http://books.google.ca/books?id=guh...t modulo 3" and "congruent modulo 4"&f=false"on googlebooks, it has the same question the answer to R1 U R2 is {(a,b):a-b is congruent to 0,3,4,6,8, or 9 (mod 12)}
My school book states the same as well.
I've been trying to figure out how they came down to this conclusion.

 

[tiny-tim]

hi nistaria!

According to this http://books.google.ca/books?id=guh...t modulo 3" and "congruent modulo 4"&f=false"on googlebooks, it has the same question the answer to R1 U R2 is {(a,b):a-b is congruent to 0,3,4,6,8, or 9 (mod 12)}

yes, i was wondering whether to point that out, and i looked at the question, and i couldn't see any reason to do so …

especially since there's no point in trying to identify equivalence classes since R₁ U R₂ simply isn't an https://www.physicsforums.com/library.php?do=view_item&itemid=151" …

personally, i think the way you wrote it is both shorter and clearer than the mod 12 way

(but maybe the rest of the question make a mod 12 answer more appropriate?)

it is 0,3,4,6,8, or 9 (mod 12) because anything divisible by 3 is 0,3,6, or 9 (mod 12), and anything divisible by 4 is 0,4 or 8 (mod 12) ​

… something for you to chew over!