Supplementary Exercises for Math 141
Exercises on logic:
1. (Selected from An Introduction to Axiomatic Systems by Burnett Meyer, Prindle, Weber and Schmidt, Boston, 1974.)
If John majors in Physics, he must study Mathematics. If he studies Mathematics, he must think logically. He is studying Mathematics.
Indicate whether each of the proposed conclusions below is a valid consequence of the paragraph above. Justify the conclusion if it is valid. Find a counterexample if it is invalid.
(a) He is majoring in Physics
(b) Either he must think logically or he is not majoring in Physics.
(c) He must think logically.
(d) He must study Mathematics only if he majors in Physics.
(e) If he majors in Physics, he must think logically.
2. (Lewis Carroll)
Nobody who really appreciates Beethoven fails to keep silence while the Moonlight Sonata is being played. Guinea pigs are hopelessly ignorant of music. No one who is hopelessly ignorant of music ever keeps silence while the Moonlight Sonata is being played.
See the instructions for Exercise 1:
(a) If one does not really appreciate Beethoven, then one is hopelessly ignorant of music.
(b) Guinea pigs do not really appreciate Beethoven.
(c) Either one does not really appreciate Beethoven or one is not a guinea pig.
(d) Being a guinea pig is a sufficient condition for one not to keep silence while the Moonlight Sonata is being played.
Problems 3-5 are selected from Mathematics: Problem Solving Through Recreational Mathematics, B. Averbach and O. Chein, Freeman, New York, 1980.
3. All Cyclopses have only one eye. Humanoids cannot get a driver's license unless they have depth perception. No one eyed creatures have depth perception. Polyphemus is a Cyclops. Cyclopses are humanoids.
Can Polyphemus get a driver's license? Justify your answer.
4. According to Lewis Carroll in The Hunting of the Snark (1876): All Boojums are snarks. Every Bandersnatch is a frumious animal. Only animals which frequently breakfast at five o'clock tea can be snarks. No frumious animals breakfast at five o'clock tea.
Are any Bandersnatches Boojums? Justify your answer.
5. Everyone who believes that mermaids exist has visited Atlantis. Only people who have found favor with the gods are protected by Neptune. No one who has not tasted real ambrosia has partaken food with the gods. People who do not believe that mermaids exist have never seen a mermaid. Everyone who has tasted real ambrosia has a hangover. Only people who have partaken food with the gods have ever been invited to Mount Olympus. No one who has not survived a shipwreck has visited Atlantis. Everyone who has found favor with the gods has been invited to Mount Olympus. No one who is not protected by Neptune has survived a shipwreck.
What, if anything, can we conclude about the connection between seeing a mermaid and having a hangover? Justify your answer.
Exercises on Sets:
1. A farmer has 41 pigs. He notices that every fat pig is greedy. Twenty of his pigs are both fat and healthy; eight of his pigs are greedy but not healthy; and thirty of his pigs are greedy. Also, one pig is neither healthy nor greedy; and five pigs are greedy but not fat. How many pigs are healthy but not greedy? How many are greedy but neither fat nor healthy? How many are healthy and greedy but not fat?
2. Using "pick-a-point proofs", prove DeMorgan's Laws for Relative Complements:
[A - (B C)] = [(A - B) (A - C)]
[A - (B C)] = [(A - B) (A - C)]
3. Construct Venn diagrams for the following set identities:
[(A B) C] = [A (B C)] Associativity of Intersection
[(A B) C] = [A (B C)] Associativity of Union
[A ( B C)] = [ (A B) (A C)] Distributivity of Intersection over Union
[A ( B C)] = [ (A B) (A C)] Distributivity of Union over Intersection
(A B)' = (A' B') Complement of Union
(A B)' = (A' B') DeMorgan's Laws Complement of Intersection
A - B = A B'
[A - (B C)] = [(A - B) (A - C)]
[A - (B C)] = [(A - B) (A - C)] DeMorgan's Laws for Relative Complements
4. Prove:
a) ( A B ) × ( C D ) = ( A × C ) ( B × D )
b) ( A \ B ) × C = ( A × C ) \ ( B × C )
5. Define the symmetric difference of two sets M and N by
M N = ( M \ N ) ( N \ M ).
a) Draw a Venn diagram for M N.
b) Prove the identity M N = ( M N ) \ ( M N ).
c) Prove that symmetric difference has the following properties:
i) L M = M L;
ii) ( L M ) N = L ( M N )
iii) M M =
iv) M = M
v) M U = M
vi) L ( M N ) = ( L M ) ( L N )
where is the empty set and U is the universal set.
Exercises on Relations:
1. For each of the relations listed below,
i) Give an example, if possible, of two distinct elements of the set which are related.
ii) Give an example, if possible, of two elements which are not related.
iii) Determine whether the relation is reflexive, symmetric, antisymmetric,transitive. For each property which does hold, it is not necessary to prove the property, but give and English statement of what it means that the property holds; foreach property which does not hold, give a counterexample.
iv) For each relation which is an equivalence relation, either describe the equivalence classes or else describe a set which is in one to one correspondence with the equivalence classes (whichever seems more appropriate).
v) For each order relation, determine whether or not it is a total ordering. If it is not a total ordering, give two elements which are not comparable.
a) Equality, on any set X; (I.e. xRy if x = y .)
b) The empty set, on any set X; (I.e., nothing is related to anything.)
c) X × X, on any set X; (I.e., everything is related to everything.)
d) "is not equal to", on the set of real numbers;
e) "is less than", on the set of real numbers;
f) "is the squareroot of", on the set of positive real numbers;
g) "is less than or equal to", on the set of real numbers;
h) "has the same sign as" on the set of real numbers;
i) "has the same square as", on the set of real numbers;
j) "has the same square as", on the set of nonnegative real numbers;
k) "has the same integer part as", on the set of real numbers;
l) "has the same fractional part as" on the set of real numbers;
m) "is a divisor of", on the set of all positive integers;
n) "is a divisor of", on the set of all integers;
o) "is a multiple of", on the set of all integers;
p) "has the same parity as", on the set of integers;
q) "is congruent to modulo n", on the set of all integers; (I.e., xRy if y-x is divisible by n,
where n is a fixed positive integer.)
r) xRy if xy > 0, on the set of real numbers;
s) xRy if xy 0, on the set of real numbers;
t) lexicographic ordering on the set of ordered pairs of real numbers; (I.e., (a,b) (c,d)
if a < c or if a = c and b d.)
u) "is at least as long as", on the set of all line segments in a plane;
v) "has the same length as", on the set of all line segments in a plane;
w) "is or is parallel to", on the set of all line segments in a plane;
x) "is or is perpendicular to", on the set of all line segments in a plane;
y) "lies on the same circle with center at the origin as", on the set of all points in a
coordinate plane;
z) "is congruent to", on the set of all triangles in a plane;
(continued on Page 4)
aa) "is similar to", on the set of all triangles in a plane;
bb) "has the same area as", on the set of all triangles in a plane;
cc) "are concentric", on the set of all circles in a plane;
dd) "are concentric and the radius of the first does not exceed the radius of the second", on the set of all circles in a plane;
ee) "has the same radius as", on the set of all circles in a plane;
ff) "is tangent to", on the set of all circles in the plane;
gg) "is or is the brother of", on the set of all people (i.e., xRy if x = y or x is the brother of y);
hh) "is or is the sibling of", on the set of all people;
ii) "is or is an ancestor of", on the set of all people;
jj) "is or is the grandfather of" on the set of all people;
kk) "is the spouse of", on the set of all people;
ll) "is the same age as", on the set of all people;
mm) "was born in the same city as", on the set of all people;
nn) "is or is an acquaintance of", on the set of all people;
oo) "had the same temperature as at noon EST today", on the set of all cities in the U.S.;
pp) "is east of", on the set of all cities in the U.S.;
qq) "is east of", on the set of all cities in the world;
rr) "logically implies", on the set of all propositional forms;
ss) "is logically equivalent to", on the set of all propositional forms;
tt) "has the same truth value as", on the set of all statements;
uu) "has a true conjunction with", on the set of all statements; (i.e., pRq if pq is true.)
vv) "has a true disjunction with", on the set of all statements;
ww) "has a true conditional with", on the set of all statements; (i.e., pRq if pq is true.)
xx) "has a true biconditional with", on the set of all statements;
yy) "has the same truth set as", on the set of all propositional functions;
zz) "is row equivalent to", on the set of m × n matrices;
ab) "has the same rank as" on the set of m × n matrices;
ac) "is similar to", on the set of n × n matrices;
ad) "is congruent to", on the set of n × n matrices;
ae) "has the same number of elements as", on the set of all subsets of a finite set X;
af) "is a subset of", on the set of all subsets of a set X;
ag) "has empty intersection with", on the set of all subsets of a set X;
ah) "is the complement of", on the set of all subsets of a set X;
ai) "has the same degree as", on the set of all polynomials
aj) "is the derivative of", on the set of all differentiable functions;
ak) "is some derivative of", on the set of all infinitely differentiable functions;
al) "is an indefinite integral of", on the set of all continuous functions;
am) "is bounded above by", on the set of all continuous functions.
an) "is the same as or lexicographically precedes", on the set of all words in the English
language.
2. Let S be a set and let T be the power set of S. Consider T as a partially ordered set, where T is ordered by set inclusion. (That is, is X and Y are subsets of S then X R Y iff
X Y.
a) Find all minimal elements of S.
b) Find all maximal elements of S.
c) Does S have a minimum element? If so, what is it?
d) Does S have a maximum element? If so, what is it?
3. Let S = (2,3,4,...} and let S × S be ordered by (m,n) (r,s) iff mr and sn.
a) Find all minimal elements of S.
b) Find all maximal elements of S.
c) Does S have a minimum element? If so, what is it?
d) Does S have a maximum element? If so, what is it?
4. State whether each of the following is true or false; if false, give a counterexample. Explain your answers.
a) If a partially ordered set has only one maximal element, then the maximal element is a
maximum element.
b) If a finite partially ordered set has only one maximal element, then the maximal element is a maximum element.
c) If a totally ordered set has only one maximal element, then the maximal element is a
maximum element.
d) If a subset of a partially ordered set has only one upper bound, then that upper bound is
the least upper bound for the subset.
e) If a subset of a partially ordered set has a least upper bound, then that least upper bound
is a maximum element for the subset.
f) If a subset of a partially ordered set has a maximum element, then that maximum element
is a least upper bound for the subset.
g) If a subset of a partially ordered set has an upper bound, then it has a least upper bound.
h) If a subset of a partially ordered set has an upper bound which is contained in the subset,
then the subset has a maximum element.
5. For each of the parts of Problem 4 which are true, state the analog for minimal elements, lower bounds, etc.
6. Let S = {p,q,r,s,t,u,v} be a partially ordered set with order relation R given by the following diagram, where an arrow or a directed path from x to y means that x R y (i.e., x precedes y, x y, etc.):
Thus, for example, t s and t p but t u.
a) Find all minimal elements of S.
b) Find all maximal elements of S.
c) Does S have a minimum element? If so, what is it?
d) Does S have a maximum element? If so, what is it?
7. Let S = {1,2,3,4,5,6,7,8} be ordered as shown in the following diagram, where an arrow or a directed path from x to y means that x R y (i.e., x precedes y, x y, etc.):
Thus, for example, 5 3, 5 2, 5 1, but 5 4.
Let A = {4,5,7}, B = {2,3,6} and C = {1,2,4,7}. For each of these subsets of S, find
a) The set of upper bounds
b) The set of lower bounds
c) A least upper bound, if one exists. (If none exists, say so.)
d) A greatest lower bound, if one exists. (If none exists, say so.)
Exercises on functions:
1. (Selected from An Introduction to Axiomatic Systems by Burnett Meyer, Prindle, Weber and Schmidt, Boston, 1974.)
Let C be the set of all citizens of Smalltown, and let H be the set of all houses in Smalltown. Let f be a relation from C to H defined by x f y iff citizen x lives in house y.
a) What assumptions are we making about the citizens of Smalltown if we say that f is a function.
b) What would it mean about Smalltown if f is not a function?
c) If f: C H is a function
i) What does it say about the housing situation in Smalltown if f is surjective?
ii) What does it say if f is injective?
iii) What does it say if f is bijective?
2. a) If S and T are subsets of A and f: A B , compare f(S T) and f(S) f(T).
b) What can be said about f(S T) ?
c) What about f-1(f(S))?
d) If K is a subset of B and if f is surjective, what can be said about f(f-1(K))?
Exercises on Algebraic Systems:
1. Determine whether or not each of the definitions of * given below gives a binary operation on the given set. If not, explain why not:
a) On N, define * by a*b = ab
b) On N, define * by a*b = c, where c is the smallest integer greater than both a and b
c) On N, define * by a*b = c, where c is at least 5 more than a+b
d) On N, define * by a*b = c, where c is the largest integer less than the product of a and b
2. Let S be a set having exactly one element. How many different binary operations could be defined on S? Answer the question if S has exactly 2 elements; exactly 3 elements; exactly n elements.
3. Let the binary operation * be defined on the set S = {a,b,c,d,e} by means of the following table:
*abcde
aabcbd
bbcaec
ccabba
dbebed
edbabc
a) Compute b*d, c*c, and [(a*c)*e]*a
b) Compute (a*b)*c and a*(b*c). Can you say on the basis of this computation whether or
not * is associative?
c) Compute (b*d)*c and b*(d*c). Can you say on the basis of this computation whether or
not * is associative?
d) Is * commutative? Why?
4. Mark each of the following as true or false:
a) If * is any binary operation on a set S, then for all a in S, a*a = a
b) If * is any commutative operation on a set S, then, for any a,b,c in S, a*(b*c) = (b*c)*a
c) If * is any associative binary operation on a set S, then, for any a,b,c in S,
a*(b*c) = (b*c)*a
d) The only binary operations of any importance are those defined on sets of numbers.
e) Every binary operation defined on a set having exactly one element is both commutative
and associative.
f) A binary operation on a set S assigns at least one element of S to each ordered pair of
elements of S
g) A binary operation on a set S assigns at most one element of S to each ordered pair of
elements of S
h) A binary operation on a set S assigns exactly one element of S to each ordered pair of
elements of S.
5. For each binary operation * defined below, determine whether * is commutative and whether * is associative.
a) On Z, define * by a*b = a-b
b) On Q, define * by a*b = ab+1
c) On Q, define * by a*b = ab/2
d) On N, define * by a*b = 2ab
e) On N, define * by a*b = ab
6. Prove that if * is an associative and commutative binary operation on a set S, then (a*b)*(c*d) = [(d*c)*a]*b, for all a,b,c,d in S.
7. For each binary operation * defined on a set below, determine whether or not * gives a
group structure of the set. If not, which group axioms fail?
a) Define * on Z by a*b = ab
b) Define * on Z by a*b = a-b
c) Define * on + by a*b = ab
d) Define * on Q by a*b = ab
e) Define * on the set of all nonzero real numbers by a*b = ab
f) define * on C by a*b = a+b
8. Each of the examples below gives a set and a binary operation defined on that set. Which of these are groups? If the example is a group, identify the identity and the inverse of each element.
a) The positive irrational numbers under multiplication
b) The set {1,-1,i,-i} under multiplication, where i2 = -1
c) The set of all rational numbers other than 1 under the operation * defined by
a*b = a+b-ab
d) The integers under subtraction
9. Construct a table for the group of symmetries of a square. (Hint: There are eight elements.)
10. Let (G,*) be a group. Prove that, for any elements a,b in G, (a*b)-1 = b-1*a-1. Find a
similar expression for (a*b*c)-1
11. Let G = {1,5,7,11}, a subset of Z12.
a) Show that (G,) is a group. [Note that (Z12,) is not a group.]
b) Is (G,) isomorphic to (Z4,+)? Justify your answer.
12. Can an abelian group be isomorphic to a non-abelian group? Justify your answer.
13. Is (Z5',) isomorphic to (Z4+)? Justify your answer. [Note (Z5',) is the group of nonzero elements of Z5 under multiplication.]
14. Prove that the composition of two isomorphisms is an isomorphism.
15. a) Prove that if (G,*) is isomorphic to (H,#) and if (H,#) is associative then (G,*) must be associative.
b) Do the same if "associative" is replaced by "commutative".
16. Prove that the image of a neutral (i.e., identity) element under an isomorphism is a
neutral element.
17. Let G be a group and let g be a fixed element of G. Define the map ig : G G by
ig(x) = g-1xg, for each x in G. Prove that ig is an automorphism of G (i.e., and
isomorphism of G with itself).
18. Let S be the set of real numbers except -1, and define * on S by a*b = a+b+ab.
a) Show that * is a binary operation on S
b) Show that (S,*) is a group
c) Show that (S,*) is isomorphic to the group (*,) of nonzero real numbers under
multiplication. (Note: You have to actually define a mapping and show it is an
isomorphism.)
19. Let X be some set and let S be the collection of all subsets of X. For A and B in S, define
two operations and by
A B = (A\B) (B\A) and A B = A B
[Note, is the same as the operation which was defined in Exercise 5 on page 2.]
a) Prove that (S,,) is a ring.
b) What is the zero element?
c) Given A in S, what is -A?
d) Is the ring commutative? Justify.
e) Is there a unity element? If so, what is it?
f) Are there divisors of 0?
g) Is (S,,) an integral domain? a field?
20. Define operations and on Z by
a b = a + b - 1 and a b = a + b - ab
a) Prove that (Z,,) is a ring.
b) What is the zero element?
c) Given a in Z, what is -a?
d) Is the ring commutative? Justify.
e) Is there a unity element? If so, what is it?
f) Are there divisors of 0?
g) Is (Z,,) an integral domain? a field?
21. Let (F,+) be a field. For a, b F, with b 0, define the operation / by a/b = a(b-1).
Prove that for any a,b,c,d F, with b 0 and d 0, a/b + c/d = (ad + bc)/(bd)
and (a/b)(c/d) = (ac)/(bd)
22. Let (R,+,) and (S,,) be isomorphic rings. Prove that if (R,+,) is a field then so is
(S,,).
23. Let F be the field of real numbers of the form a + b, where a and b are rational. Let
M M2,2(Q) be the ring of 2 × 2 rational matrices of the form a 2b Define
b a
:FM by (a+b) = a 2b
b a .
Prove that :FN is a ring isomorphism (and hence that N is a field).
24. Let be the field of real numbers and let M = M2,2() be the ring of 2 × 2 rational matrices. Define :FM by
(a+b) = a 2b
b a
Prove that is a ring isomorphism (and hence that M is a field).