Discrete Mathematics
$30-250 USD
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1. Show that (p ↔ q) and (p Ù q) Ú (Øp Ù Øq) are equivalent without using truth
table. [10 points]
2. Draw a relevant conclusion from the set of premises given below. Explain
each rule of inference you have used. [10 points]
“I am either dreaming or hallucinating.” “I am not dreaming.” “If I am
hallucinating, I see elephants running down the road.”
3. Let F(x, y) be the statement “x can fool y”, where domain consists of all
people in the world. Use quantifiers to express “No one can fool both Fred
and Jerry” [10 points]
4. Use proof by cases to show that min(a, min(b, c)) = min(min(a, b), c)
whenever a, b, and c are real numbers. [10 points]
5. Find an explicit formula for a sequence 1, 10, 11, 100, 101, 110, 111, 1000,
1001, 1010, 1011, …. [10 points]
6. Find tight big-Oh estimate of x4 + x2 logx + 2x [5 points]
7. Show that if n is an integer and n3 +5 is odd, then n is even using a) a proof by
contradiction and b) indirect proof. [10 points]
8. Show that the relation R on the set of ordered pairs of positive integers given
by ((a, b), (c, d)) Î R if and only of ad = bc, is an equivalent relation. [5
points]
9. Prove that for every integer n, 1.2.3 + 2.3.4 + … + n.(n + 1).(n + 2) = n.(n +
1).(n + 2).(n+3)/4 [5 points]
10. Find the number of derangements on the set of 15 elements. Show all the
calculations. [5 points]
11. Answer the questions for the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, | )
[20 points]
a. Find the maximal and minimal elements
b. Is there a greatest element? What about the least element?
c. Find all upper bounds of {6, 27}, also find the least upper bound if
exists.
d. Find all lower bounds of {36, 48}, also find the greatest lower bound if
exists.
项目ID： #1327060
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