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An exam for EECS 203 course taken in Spring 2015. It consists of 9 problems covering topics such as logic and sets, growth of functions and infinite sets, number theory, and proof by induction. The exam has a total of 100 points and students are allowed to use one page of notes. The document also includes instructions, honor code, and space for students to write their name and uniqname.
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(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)
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)
(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)
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.
for (i:=1 to n) x=1; while(x<n) x=x*2;
This algorithm has a run time of
Use induction to prove Prove that 3 divides n^3 + 2n whenever n is a positive integer.
Theorem:
Proof:
Base case:
Induction step:
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]
a. Find a bijection from ๏+^ to ๏.
b. Prove that the product of any three consecutive integers is divisible by 6.