MTH 250 – Graded Assignment 1
Work each of the following problems. Please organize work neatly – if you’re typing, type into
the existing Word doc. If you’re handwriting, I’ve left some space before creating the .pdf, but
one approach I appreciate is if you modify the spacing in the Word doc and print it so you can fit
your answers into the existing pages. Or, work on separate sheets (just make sure it’s well
labeled and easy to follow!).
Sets and set notation>> Basic notation and definitions
P1: Let A  {3, a, b,12, q}, B  {a, b, d ,12, q, 7}, C  {4, b, c, a, 25}
a) True or false: | C || B | .
b) True or false: The sets A and C are equipollent.
c) True or false: The sets B and C are equipollent and equal.
d) Construct a mapping f : A  C (explicitly using the sets A and C given above) which is
neither surjective nor injective.
e) Explain why any mapping from A to C which is surjective will also be injective, and vice
versa.
f)
Construct a mapping f : A  B (explicitly using the sets A and B given above) which is
injective but not surjective.
g) Is it possible to have a bijection from A to B ? Explain.
Sets and set notation>> Subsets
P2: Let A  {3, a, b,12, 0}, B  {a, b, c, 0, 4}, C  {4, b, c, a}
a) True or false: B  A .
b) True or false: C  B and C  B .
c) How many subsets does A have? How many are proper?
d) List all the subsets of C which contain three elements.
Sets and set notation>> Set operations
P3: Given the sets
X  {a, c, e, g}
Y  {a, b, c}
Z  {b, c, d , e, f }
in universe
U  {a, b, c, d , e, f , g}
find the sets given by the expressions below. Be sure to show intermediate steps.
a) ( X ' Y )  Z
b) (Y \ X )  Z '
Sets and set notation>> Venn diagrams
P4: For sets A, B, C in a universe U in general (i.e. not any of the previously given sets in
particular, but any A, B, and C ), construct a Venn diagram and shade the region
corresponding to ( A ' C )  B .


Sets and set notation>> Sets of real numbers
P5:
a) Sketch the set {x | x  ,  3  x  6} on a number line and enumerate the elements
(express the set in listing notation).
b) Sketch the set {x | x  , x < 2 or 6  x  10} on a number line and express using
interval notation.
c) Sketch the region in the plane described by {( x, y ) | y  x  1}  {( x, y ) | y  1} .
Abstract algebra>> Introduction to groups
P6: Which of the group properties (closure, associative, identity, inverse) are satisfied by
 A,*  , where A  {a, b, c} and * is defined by the Cayley table given below? Which
properties fail? Explain – each property must have some justification/explanation as to what
you’re looking at that supports your answer.
Logic and proof>> Statements and operators
P7: Let p= "birds fly", q = "dogs howl", r = "elephants are pink".
a) Translate the compound statement into symbols using the logical operators , ,  :
It is not the case that birds do not fly and dogs howl, or elephants are pink.
b) Translate into English: p  (r  q)
Logic and proof>> Truth tables
P8: Construct a truth table for the expression
q  ( p  r )
p
q
r
q
pr
q  ( p  r )
Logic and proof>> The conditional
P9: Write the inverse, converse and contrapositive of the statement
" 1  1  4 " implies " 2  5  7 "
What are the truth values of the original statement and its rearrangements?
Logic and proof>> Arguments
P10: Determine whether the arguments are valid or invalid and cite the form of the syllogism /
fallacy (show translation to symbolic form):
a) Either the sun is shining or the sky is blue.
The sun is not shining.
The sky is blue.
b) If my candidate wins the election, your taxes will be lower.
Your taxes were lowered by the winning candidate.
My candidate must have won.