6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2,
5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by
you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f
is not a function? (b) In how many ways can you pick x ∈ A and y ∈
B such that f : A → B...
Let X = {1, 2, 3, 4, 5, 6} and let ∼ be given by {(1, 1),(2,
2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 3),(1, 5),(2, 4),(3, 1),(3, 5),
(4, 2),(5, 1),(5, 3)}.
Is ∼ an equivalence relation? If yes, write down X/ ∼ .
Let F(x) = log5x and H(x) = x . Find the value of
F(H(25)).
Write the equation of a circle whose center is located at (-2,
-5) with a radius of 11 inches.
Given a triangle with Angle A = 45 degrees, Angle B = 60
degrees and side a = 10 inches.
Find the length of side b. (Leave answer in radical
form.)
Find the distance between a set of points whose coordinates
are A(-7, 8) and B(5, -11)....
Let S = {1, 2, 3, 4, 5, 6, 7} be a sample of an experiment and
let X = {1, 4, 7}, Y = {2, 3, 5}, and Z = {1, 3, 5} be events.
Which of the following statements is correct?
a) X and S are mutually exclusive events.
b) X and Y are mutually exclusive events.
c) X, Y, and Z are mutually exclusive events.
d) Z and Y are mutually exclusive events.
e) X and Z...
Let G be an abelian group.
(a) If H = {x ∈ G| |x| is odd}, prove that H is a subgroup of G.
(b) If K = {x ∈ G| |x| = 1 or is even}, must K be a subgroup of G?
(Give a proof or counterexample.)
Consider G = (Z12, +). Let H = {0, 3, 6, 9}.
a. Show that H is a subgroup of G.
b. Find all the cosets of H in G and denote this set by G/H.
[Note: If x ∈ G then H +12 [x]12 = {[h +
x]12?? | [h]12 ∈ H} is the coset generated by
x.]
c. For H +12 [x]12, H +12
[y]12 ∈ G/H define
(H+12[x]12)⊕(H+12[y]12)
by(H+12 [x]12)⊕(H+12
[y]12)=H+12 [x+y]12.
d. Show that ⊕ is...
4) Let ? = {2, 3, 5, 7}, ? = {3, 5, 7}, ? = {1, 7}. Answer the
following questions, giving reasons for your answers.
a) Is ? ⊆ ??
b)Is ? ⊆ ??
c) Is ? ⊂ ??
d) Is ? ⊆ ??
e) Is ? ⊆ ??
5) Let ? = {1, 3, 4} and ? = {2, 3, 6}. Use set-roster notation
to write each of the following sets, and indicate the number of
elements in...