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Literally the first homework question, and I seem to be struggling. There doesn't seem to be any examples in our book, so I'm hoping someone might help walk me through it. I'm guessing it's pretty simple too...

I'm not looking for the answer, I'd just like some pointers on what I need to do, and how I should go about enumerating the elements.

  1. Enumerate the elements of the following relations from the set A of positive integers less than or equal to 10 to the set B of positive integers less than or equal to 30.

    1. An element a of A is related to the element b of B if b = 3 × a

      R=(1,3)(2,6)(3,9)(4,12)(5,15)(6,18)7,21)(8,24)(9,27)(10,30)

    2. An element a of A is related to the element b of B if b = 2 × a - 1

      R=(1,1)(2,3)(3,5)(4,7)(5,9)(6,11)(7,13)(8,15)(9,17)(10,19)

At first I thought I'd just have the two sets: A = {1,2 3 ,4 5 ... 10 } B = {1, 2, 3, 4, 5 ... 30 }

so would I just take a (lets say 1) and b (1 again) 1 = 3 * 1 (obviously false)? and continue on for each element in both sets? 3 = 3 * 1 (true)

It just seems a little tedious to go through all combinations to test.

Rickybobby
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  • $3a$ and $2a - 1$ are one-to-one functions so there is exactly one related element for each of $a \in {1, 2, ..., 10}$ (all of which will be in $B$). And vice versa, for each element in $B$ there is exactly one (or none) elements in $A$. – Jared Jun 25 '16 at 03:12
  • Notice that $3A = {3 \times a : a \in A} = {3, 6, 9, .., 30} \subseteq B$. – Steven Harding Jun 25 '16 at 03:13

2 Answers2

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The notation your book uses may be different, but it is common to denote a relation by something like "$aRb$" to represent that $a$ is related to $b$.

The first set of relations would look like $1R3$, $2R6$, etc.

Saying that two things are related is not the same as saying they're equal -- there are equivalence relations, which behave similarly to the standard "$=$", but there are plenty of other relations that do not; for instance, the relation "$<$" on $\mathbb{N}$. A relation is just a set of ordered pairs, so it does not even need to have a simple English definition.

Also, it is important to note that by default, relations are not symmetric; it is true that in your first example, we have $1R3$, but we do not have $3R1$. Nor are they necessarily reflexive: here, $1$ is not related to $1$.

These should be comments, but I lack the reputation required to do that.

jammminy
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  • So what you're saying for number one it would be (1,3) and (2,6)? or am I misunderstanding? – Rickybobby Jun 25 '16 at 03:11
  • $(1,3), (2,6), \ldots, (10,30)$, if you want to write your relations in that way. – jammminy Jun 25 '16 at 03:14
  • or 1R3 ... 2R6 like you mentioned – Rickybobby Jun 25 '16 at 03:14
  • @beatles1235 Yes, but I recommend following the notation of your textbook; it doesn't make any logical difference which notation you use, but I assume they will use a consistent notation throughout the book. – jammminy Jun 25 '16 at 03:16
  • Got it! I honestly think that the book we're using is terrible. very little discriptions, barely any examples. Any books you would recommend? or should I just contact the professor? – Rickybobby Jun 25 '16 at 03:17
  • @beatles1235 Sorry, I don't have a favorite set theory book at this level; you could try asking in chat. I also think I remember there being a question about that on this site. – jammminy Jun 25 '16 at 03:21
  • Got it, well I'll work on 1. and post my answer when I'm done! I think i've got a better understanding – Rickybobby Jun 25 '16 at 03:22
  • For 1, it looks like R=(1,3)(2,6)(3,9)(4,12)(5,15)(6,18)7,21)(8,24)(9,27)(10,30) – Rickybobby Jun 25 '16 at 17:15
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A binary relation between sets $A$ and $B$ is a subset of the cartesian product $A \times B :=\{(a,b): a \in A, b \in B\}$. In your case, the set $A \times B$ contains 300 elements. The first relation is the subset $\{ (a, 3a): a \in A\} = \{ (1,3), (2,6), \ldots, (10,30) \}$ consisting of 10 elements. The second relation consists of the 10 elements $(1, 1), (2, 3), (3, 5), \ldots, (10, 19)$.

One can also define binary relations on a single set $A$ to be any subset of $A \times A$.

Ashwin Ganesan
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