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EXAM 1โmakeup
EECS 203
Spring 2015
Name (Print): ________________________________________________
uniqname (Print): ________________________________________________
Instructions. You have 110 minutes to complete this exam. You may have one page of notes
(8.5x11.5 two-sided) but may not use any other sources of information, including electronic
devices, textbooks, or notes. Leave at least one seat between yourself and other students. Please
write clearly. If we cannot read your writing, it will not be graded.
Honor Code. This course operates under the rules of the College of Engineering Honor Code. Your
signature endorses the pledge below. After you finish your exam, please sign on the line below:
I have neither given nor received aid on this examination, nor have I concealed any
violations of the Honor Code.
________________________________________________________________
Problem # Points
Total /
- Logic and sets (15 points)
In this section, each question will have zero or more correct answers. You are to circle each
correct answer and leave uncircled each incorrect answer.
[3 points each, -1 per incorrect circle/non-circle, minimum 0 points per problem]
a) Circle each of the following that is a tautology.
(๐ โ ๐) โ (ยฌ๐ โ a)
(๐ โ ๐) โ (ยฌ๐ โ ยฌ๐)
(๐ โจ ๐) โ ๐
(๐ โง ๐ โง ๐) โ (๐ โจ ๐ โจ ๐)
b) Circle each of the following that is satisfiable.
c) Circle each of the following which are tautologies.
(๐จ โ (๐จ โฉ ๐ฉ) โ (|๐ฉ| = |๐จ|))
|๐จ โช ๐ฉ| โฅ |๐จ| + |๐ฉ|
((๐จ โ ๐ฉ) = {๐}) = (|๐จ| = |๐ฉ|)
(๐จ โ ๐ฉ = {๐}) โ ((๐ โ ๐จ) โจ (๐ โ ๐) )
( ๐จ โ ๐ช = ๐ฉ โ ๐ช) โ (๐จ = ๐ฉ)
- More Sets (4 points)
(No partial credit will be given on this problem.)
Shade the Venn diagram below to show (๐ต โช ๐ด) โฉ (๐ถ โช ๐ต) โฉ (๐ต โช ๐ถ)
- Functions (6 points)
In this section, each question will have zero or more correct answers. You are to circle each
correct answer and leave uncircled each incorrect answer.
[3 points each, -1 per incorrect circle/non-circle, minimum 0 points per problem]
a) Say that f is a function from ๐ด โ ๐ต where A and B are both subsets of ๏ (the natural numbers). If |B|>|A| then you can conclude that
f is not onto
f is not one-to-one
f is a bijection
f is not a bijection
The cardinality of A is greater than or equal to the cardinality of the range of f.
b) Say that f is a function from ๐ด โ ๐ต where A and B are both subsets of ๏ (the natural numbers). If B ๏ A then you can conclude that
f is not onto
f is not one-to-one
f is a bijection
f is not a bijection
The cardinality of A is greater than or equal to the cardinality of the range of f.
- Growth of functions and infinite sets (10 points)
In this section, each question will have zero or more correct answers. You are to circle each
correct answer and leave uncircled each incorrect answer.
Each problem is worth 2 points and you only get the points if you circle all of the correct
answers and none of the wrong ones.
a) log(X)+X is:
๏ (X) O(X^2 ) ๏ (X) ๏ (log(X)) O(log(X))
b) Consider the following pseudo code:
for (i:=1 to n) x=1; while(x<n) x=x*2;
This algorithm has a run time of
๏ (n) ๏ (n^2 ) ๏ (log(n)) O(n^3 ) O(n/2+1)
c) X^4 +12X^2 log(X)+X is:
๏ (X^4 ) O(X^4 ) ๏ (X^4 ) ๏ (X^3 ) ๏ (X^3 )
d) If A and B are both countably infinite sets, then ๐ด โ ๐ต could be
Countably infinite Uncountably infinite Finite
e) If A and B are both uncountably infinite sets, then ๐ด โช ๐ต could be
Countably infinite Uncountably infinite Finite
- Proof by induction (12 points)
Use induction to prove Prove that 3 divides n^3 + 2n whenever n is a positive integer.
Theorem:
Proof:
Base case:
Induction step:
- Rules of logic (16 points)
Prove the following logical equivalence (in two different ways):
(๐ โ (๐ โ ๐)) โก ยฌ(๐ โง ๐ โง ยฌ๐)
(a) Use a truth table [6]
(b) Write a formal proof, justifying each line [10]
- Other techniques (12 points, 6 each)
a. Find a bijection from ๏+^ to ๏.
b. Prove that the product of any three consecutive integers is divisible by 6.
