Baby Exam
M366
Spring 2012
Name:
Score:
Read the problems. Read them again. Make sure to express your answers
in a clear and logical way. Don’t rush!
1. (10 pts.) Let
A = {1, 5, 7, f ish, cow, red, U SA}, B = {1, 4, 7, f ish, kebab}.
Write down:
(a)
A ∩ B.
(b)
Ac ∩ B.
2. (15 pts.) Give an example of:
(a) an injective function.
(b) a function which is surjective but not injective.
(c) a function which is neither surjective nor injective.
Note! Remember to specify which are the initial and the arrival sets of
your functions!!!
3. (25 pts.) TRUE-FALSE: no explanation is needed for this problem:
(a) For any set A,
A∪∅=∅
(b) For any set A,
A⊆A∪B
(c) If
A ∪ B = A ∩ B,
then A = B.
(d) Let f : X → Y and g : Y → Z be surjective functions. Then the
composite function
g◦f :X →Z
is a surjective function.
(e) Consider the function f : Z → Z defined by
f (n) = n + 5.
f is a bijective function.
(f) Consider the function f : N → N defined by
f (n) = n + 5.
f is an injective function.
(g) Consider the function f : N → N defined by
f (n) = n + 5.
f is a surjective function.
4. (20 pts.) Let X = {1, 2, 3, 4, 5} and Y = {y, u, p}. Consider f : X → Y
defined as follows:
f (1) = p,
f (2) = u,
f (3) = p,
f (4) = p,
f (5) = y.
(a) is f a function from X to Y ?
(b) is it injective/surjective/bijective?
(c) if f is a surjective function, write down what the equivalence classes
induced by f on X are.
5. (20 pts.)
Let X = {1, 5, 18, 37, 121, 234, 1001}.
Define an equivalence relation on X as follows: two elements of X are
equivalent if they have the same number of digits.
Comments:
You don’t have to prove that this rule is an equivalence relation. You can
assume I gave you a good one!
(a) How many elements does the quotient set X/ ∼ have?
(b) Write down the equivalence classes for this equivalence relation.
(c) Write down the partition that this equivalence relation determines
for the set X.
(d) Write down explicitly the projection function p : X → X/ ∼.
6. (10 pts.) Prove that if X and Y are finite sets with the same number of
elements and
f :X→Y
is a function from X to Y , then f is injective if and only if it is surjective.
Note: if you cannot write down a nice and streamlined proof of this, try
to expose your thoughts on it. But please, please, please, do read over your
sentences and make sure they make sense (as english sentences I mean!)