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1.Show that each of these conditional statements is a tautology by using truth tables.
a) (p∧q)→ p
b) p → (p∨q)
c) ¬p → (p → q)
d) (p∧q)→ (p → q)
e) ¬(p → q)→ p
f) ¬(p → q)→¬q
2.Refer to 1, Show that each of these conditional statements is a tautology by using Propositional Logic
Question 1: Draw a graph with the specified properties or give reason to show that no such graph exists.
i. A graph with four vertices of degree 1,1,2 and 3
ii. A graph with four vertices of degree 1,1,3 and 3
iii. A simple graph with four vertices of degree 1,1,3 and 3
Show that the explicit sequence {an} where an= 2n+1
-1 for n > or = to 1 is a
solution of the recurrence relation: an= 3an-1 – 2an-2 , n >or = to 3
6 divides n^3 - n for n>=2
Prove by mathematical induction: 6 divides n^3 - n for n>=2
Let A = {a, b, {b}, {b, e}, e} and B = {b, {b, e}, e, f}.
A ⋂ B = {b, {b, e}, e}
a. false
b. true
From a committee of 10 people,
1. In how many ways can we choose a chairperson, a vice-chairperson, and a secretary, assuming that one person cannot hold more than one position?
2. In how many ways can we choose a subcommittee of 3 people?
3. Find the number of combinations of 13 objects taken 8 at a time.
4. How many 5-card hands will have 3 aces and 2 kings?
5. Serial numbers for a product are to be made using 2 letters followed by 3 numbers. If the letters are to be taken from the first 8 letters of the alphabet with no repeats and the numbers are taken from the 10 digits (0-9) with no repeats, how many serial numbers are possible?
6. A company has 7 senior and 5 junior officers. An ad hoc legislative committee is to be formed. In how many ways can a 4-officer committee be formed so that it is composed of
a. Any 4 officers?
b. 4 senior officers?
c. 3 senior officers and 1 junior officer?
d. 2 senior officers and 2 junior officers?
e. At least 2 senior officers?
Find the value of a2 for the recurrence relation an=17an-1+30n, where a0=3
Find the value of a3 for the recurrence relation an=17an-1+30n, where a0=3
Find the value of a1 for the recurrence relation an=17an-1+30n, where a0=3
What would be the hypothesis of the mathematical induction for x(x + 1) < x! , where x ≥ 7?
If P(k) = k2(k + 2)(k – 1) is true, then what is P (k + 1)?
A. Determine if 1781 is divisible by 3, 6, 7, 8, and 9. (5 items x 2 points)
B. Determine if each of the following numbers is a prime or composite.
6. 828
7. 1666
8. 1781
9. 1125
10. 107
C. Find the greatest common divisor of each of the following pairs of integers.
11. 60 and 100
12. 45 and 33
13. 34 and 58
14. 77 and 128
15. 98 and 273
D. Find the least common multiple of each of the following pairs of integers.
16. 72 and 108
17. 175 and 245
18. 150 and 70
19. 32 and 27
20. 540 and 504
At the beginning of the first day (day 1) after grape harvesting is completed, a grape grower has 8000 kg of grapes in storage. On day n, for n = 1, 2, . . . , the grape grower sells 250n/(n + 1) kg of the grapes at the local market at the price of $2.50 per kg. He leaves the rest of the grapes in storage where each day they dry out a little so that their weight decreases by 3%. Let wn be the weight (in kg) of the stored grapes at the beginning of day n for n ≥ 1 (before he takes any to the market).
(a) Find the value of wn for n = 2.
(b) Find a recursive definition for wn. (You may find it helpful to draw a timeline.)
(c) Let rn be the total revenue (in dollars) earned from the stored grapes from the beginning of day 1 up to the beginning of day n for n ≥ 1. Find a recursive formula for rn.
