Question

Let S = {a,b,c,d,e,f,g} and let T = {1,2,3,4,5,6,7,8}.

a. How many diﬀerent functions are there from S to
T?

b. How many diﬀerent one-to-one functions are there from S to
T?

c. How many diﬀerent one-to-one functions are there from T to
S?

d. How many diﬀerent onto functions are there from T to
S?

Answer #1

and

a) Number of diﬀerent functions are there from S to T are

b) Number of diﬀerent one-to-one functions are there from S to T are

c) Number of diﬀerent one-to-one functions are there from T to S are

because ,

d) Number of diﬀerent onto functions are there from T to S are

We have 1,2,3,4,5,6,7,8,a,b,c,d,e,f
the numbers and letters are in order. How many permutations
(arrangements) are there in a way that the order is maintained for
numbers and the order is maintained for letters? Explain.

Let z=f(a,b,c) where a=g(s,t), b=h(l(s+t),t), c=tsin(s). f,g,h,l
are all differentiable functions. Compute the partial derivatives
of z with respect to s and the partial of z with respect to t.

Let f and g be continuous functions from C to C and let D be a
dense
subset of C, i.e., the closure of D equals to C. Prove that if
f(z) = g(z) for
all x element of D, then f = g on C.

9. Let S = {a,b,c,d,e,f,g,h,i,j}.
a. is {{a}, {b, c}, {e, g}, {h, i, j}} a partition of S?
Explain.
b. is {{a, b}, {c, d}, {e, f}, {g, h}, {h, i, j}} a partition
of S? Explain. c. is {{a, b}, {c, d}, {e, f}, {g, h}, {i, j}} a
partition of S? Explain.

let
A = { a, b, c, d , e, f, g} B = { d, e , f , g}
and C ={ a, b, c, d}
find :
(B n C)’
B’
B n C
(B U C) ‘

1. Let A = {1,2,3,4} and let F be the set of all functions f
from A to A. Prove or disprove each of the following
statements.
(a)For all functions f, g, h∈F, if f◦g=f◦h then g=h.
(b)For all functions f, g, h∈F, iff◦g=f◦h and f is one-to-one
then g=h.
(c) For all functions f, g, h ∈ F , if g ◦ f = h ◦ f then g =
h.
(d) For all functions f, g, h ∈...

There are two sets S and T. |S|=4 and |T|=10, how many one to
one functions and onto functions can be made from these sets?

Let S = {a, b, c, d, e, f} with P(b) = 0.21, P(c) = 0.11, P(d) =
0.11, P(e) = 0.18, and P(f) = 0.19. Let E = {b, c, f} and F = {b,
d, e, f}. Find P(a), P(E), and P(F).

Let A, B, C be sets and let f : A → B and g : f (A) → C be
one-to-one functions. Prove that their composition g ◦ f , defined
by g ◦ f (x) = g(f (x)), is also one-to-one.

2. Let A = {a,b} and B = {1,2,3}.
(a) Write out all functions f : A → B using two-line notation.
How many diﬀerent functions are there, and why does this number
make sense? (You might want to consider the multiplicative
principle here).
(b) How many of the functions are injective? How many are
surjective? Identify these (circle/square the functions in your
list).
(c) Based on your work above, and what you know about the
multiplicative principle, how many...

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