Equivalence relations and partitions.
- Thread starter tyuip
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What is your question? This problem looks extremely straightforward.
The equivalence relation is defined like this:
\(\displaystyle 1\equiv_A 3 \equiv_A 6\)
\(\displaystyle 2\equiv_A 7 \equiv_A 8\)
\(\displaystyle 4 \equiv_A 5 \equiv_A 9 \equiv_A 10\)
The function has the following definition:
\(\displaystyle f(1) = f(3) = f(6) = A_1\)
\(\displaystyle f(2) = f(7) = f(8) = A_2\)
\(\displaystyle f(4) = f(5) = f(9) = f(10) = A_3\)
What are you having trouble with?
The equivalence relation is defined like this:
\(\displaystyle 1\equiv_A 3 \equiv_A 6\)
\(\displaystyle 2\equiv_A 7 \equiv_A 8\)
\(\displaystyle 4 \equiv_A 5 \equiv_A 9 \equiv_A 10\)
The function has the following definition:
\(\displaystyle f(1) = f(3) = f(6) = A_1\)
\(\displaystyle f(2) = f(7) = f(8) = A_2\)
\(\displaystyle f(4) = f(5) = f(9) = f(10) = A_3\)
What are you having trouble with?
A relation in a set of ordered pairs. So the first answer should be a set of pairs: \(\displaystyle R = \bigcup\limits_{k = 1}^3 {\left( {{A_k} \times {A_k}} \right)} \).
Note that the equivalence relation $R$ contains $9+9+16=34$ pairs.
Don't feel bad about that. Over a period of forty years, I used more that twenty different textbooks to teach this material. Although many of those authors used some of the same notations, I dare say that no two were exactly alike.
So what you must do is learn the notation is your one textbook and stick with it.
In my view, it is impossible to learn this material from multiple sources.
So would i simply write out all the ordered pairs? Like does this seem like a proper answer?
(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)(8,8)(9,9)(10,10)(1,3)(3,1)(1,6)(6,1)(3,6)(6,3)(2,7)(7,2)(7,8)(8,7)(2,8)(8,2)
(4,5)(5,4)(4,9)(9,4)(4,10)(10,4)(5,9)(9,5)(5,10)(10,5)(9,10)(10,9)
(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)(8,8)(9,9)(10,10)(1,3)(3,1)(1,6)(6,1)(3,6)(6,3)(2,7)(7,2)(7,8)(8,7)(2,8)(8,2)
(4,5)(5,4)(4,9)(9,4)(4,10)(10,4)(5,9)(9,5)(5,10)(10,5)(9,10)(10,9)
This is simply a matter of "Do you understand how an equivalence relation 'induces' a partition?}. The partition induced on a set by an equivalence relation is just the collection of subsets in which all members that are equivalent to one another are in the same subset. Since one subset in the partition is {1, 3, 6}, we must have 1≡A 3, 1≡A 6, 3≡A 6. Since another subset is {2, 7, 8} we have ....