Let A = {1,2,3,4,5,6}, and consider the following equivalence relation on A: ! = {(1,1), (2, 2), (3, 3), (4, 4), (5,5), (6,6), (2, 3), (3,2), (4, 5), (5, 4), (4, 6), (6,4), (5, 6), (6,5)}. R = List the equivalence classes of R.
1. Let A = {1,2,3,4,5,6}, and consider the following equivalence relation on A: ! = {(1,1), (2, 2), (3, 3), (4, 4), (5,5), (6,6), (2, 3), (3,2), (4, 5), (5, 4), (4, 6), (6,4), (5, 6), (6,5)}. R = List the equivalence classes of R.
2. Let A = {a,b,c,d,e}. Suppose R is an equivalence relation on A. Suppose R has two equivalence classes. Also aRd, bRc and eRd. Write out R as a set.
3. Let A = {a,b,c,d,e}. Suppose R is an equivalence relation on A. Suppose R has three equivalence classes. Also aRd and bRc. Write out R as a set.
4. Let A = {a,b,c,d,e}. Suppose R is an equivalence relation on A. Suppose also that aRd and bRc, eRa and cRe. How many equivalence classes does R have?
5. There are two different equivalence relations on the set A = {a, b}. Describe them. Diagrams will suffice.
6. There are five different equivalence relations on the set A = {a,b,c}. Describe them all. Diagrams will suffice.
7. Define a relation R on Z as xRy if and only if 3x-5y is even. Prove R is an equivalence relation. Describe its equivalence classes.
8. Define a relation R on Z as xRy if and only if x² + y² is even. Prove R is an equivalence relation. Describe its equivalence classes.
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