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.
Let f(n) = sqrt(4n), and let g(n) = lg 4n.
Determine whether f is O(g) and whether
g is O(f).
(Notation: sqrt(w) is the square root of w.
And lg n is the logarithm of n
to the base 2.)
4.
Assume that E is an equivalence relation on a set S.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Enderson
 Math, LG, Equivalence relation, Transitive relation

Click to edit the document details