Binary Relations

[devil2euz]

Can somebody explain me how to solve this problem?

Let the R relation be A = {1,2,3,4}
aRb, if 3a+5b is a prime number

Define the set R

Define the set {x | xR2}

 

[pka]

Can somebody explain me how to solve this problem?

Can you post a verification that each of these pairs belong to \(\mathcal{R}\)?
\((2,1),~(1,4),~(4,1),~(1,2),~(3,2),~(3,4),~(2,3)\)
There is one that does not belong which is it. Are there others left out?

 

[Steven G]

You look at ever pair of numbers in {1, 2, 3, 4} and see if they are related. You just have to get your hands dirty.

 

[pka]

You look at ever pair of numbers in {1, 2, 3, 4} and see if they are related. You just have to get your hands dirty.

Actually it is not each pair, but each ordered pair,
There are \(\dbinom{4}{2}=6\) pairs, but there \(4\times 4=16\) ordered pairs.

 

[devil2euz]

Can you post a verification that each of these pairs belong to \(\mathcal{R}\)?
\((2,1),~(1,4),~(4,1),~(1,2),~(3,2),~(3,4),~(2,3)\)
There is one that does not belong which is it. Are there others left out?

The first one is ok. (2,3) doesn't belong to the relation.
How should I start the second one : {x | xR2} ?

 

[HallsofIvy]

The ordered pairs from A= {1, 2, 3, 4} are {(1, 1), (1, 2). (1, 3), (1, 4). (2, 1). (2, 2). (2, 3), (2, 4). (3, 1), (3, 2), (3, 3), (3, 4). (4. 1), (4, 2), (4, 3), (4, 4)}. As pka said there are 16 of those- 4 that start with 1, 4 that start with 2, 4 that start with 3. and 4 that star with 4.

The "relation" "3a+ 5b is a prime number" consists of all those pairs, (a. b), that satisfy. The only way to do this is to actually calculate 3a+ 5b for each pair.

(1,1): 3(1)+ 5(1)= 8. That is not a prime number so (1, 1) is not in the relation.
(1,2); 3(1)+ 5(2)= 13. That is a prime number so (1, 2) is in the relation.
(1,3): 3(1)+5(3)= 18. That is not a prime number so (1, 3) is not in the relation.

Can you complete this?

 

[devil2euz]

Can you complete this?

Yes I can, but I don't know how to deal with this: {x | xR2} . What is it even mean?

 

[Dr.Peterson]

Can you complete this?

Yes I can, but I don't know how to deal with this: {x | xR2} . What is it even mean?

It is asking for the set of elements x in A such that x R 2, that is x is related to 2 by the relation R.

In other words, for what numbers x in {1, 2, 3, 4} is 3(x)+5(2) a prime number?

 

[devil2euz]

Now it's clear. Thank you!