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Practice First Test
Mathematics 61
Disclaimer: Listed here is a selection of the many possible sorts of problems.
The
actual test is apt to make a different selection.
1.
Let S5 be the set of all binary strings of length 5 (such as 11001).
Let E be the
equivalence relation on S5 for which sEt if and only if s and t have the same first three
bits.
(For example, 11011 E 11010.)
(a)
Find [10101], the equivalence class of the string 10101.
(b)
How many equivalence classes are there altogether?
(Support your answer.)
(c)
Let A = {4, 5, 6, 7} and let P = {{5, 7}, {4, 6}}.
Then P is a partition of A.
Give the matrix (relative to numerical order on A) of the equivalence relation R on A
for which A/R = P.
2.
(a)
Assume that R is a transitive relation.
Show that whenever (x, y) is in R o R,
then (x, y) is also in R.
(b)
Assume that Q is a relation and that Q o Q is a subset of Q.
Prove that Q is
transitive.
3.
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This note was uploaded on 03/20/2008 for the course MATH 61 taught by Professor Enderson during the Fall '08 term at UCLA.
 Fall '08
 Enderson
 Math

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