Search & Filtering
p ∧¬q
Example 1:
Let P, Q and R be the propositions
P: It’s raining outside.
Q: It’s safe to drive.
R: The roads are slippery.
a) Though it’s raining outside but the roads are not slippery.
b) It’s safe to drive if and only if the roads are not slippery:
c) If it’s raining outside and the roads are not slippery then it’s safe to drive.
d) Driving is safe after the rain stopped if and only if the roads are not slippery.
Using the handshaking principle,determine the number edges of a graph with fourteen vertices and each with degree six
How many bit strings of length 11 have more 0s than 1s?
5. If P(k) = k2 (k + 2)(k – 1) is true, then what is P (k + 1)?
A. (k + 1)2 (k + 2)(k)
B. (k + 1)2 (k + 2)(k)
C. (k + 1)(k + 3)(k)
D. (k + 1)2 (k + 3)(k)
6. Using the principle of mathematical induction, 2n-1 is divisible by which of
the following?
A. 1
B. 0
C. 4
D. ½
7. A relation represents an equation where the next term is dependent on the
previous term is called
A. Binomial relation
B. Recurrence relation
C. Regression relation
D. None of these
8. Calculate the value of a2 for the recurrence relation an=17an-1+30n, where
a0=3. (2 pts)
A. 2346
B. 1296
C. 1437
D. 5484
9. The recurrence relation for Fibonacci sequence is
A. Fn = Fn + 1
+ Fn - 2
B. Fn = Fn - 1
+ Fn - 2
C. Fn = Fn - 1
- Fn - 2
D. None of these
10. In recurrence relation, a0 represents
A. Current value
B. Starting value
C. The value of next term in the sequence
D. None of these
MATHEMATICAL INDUCTION AND RECURRENCE
1. What is the base case for inequality 3n > n2 , where n = 2? (2 pts)
A. 3 > 1
B. 9 > 4
C. 6 > 4
D. 4 < 9
2.For the mathematical induction to be true, what type of number should be the value of n?
A. natural number B. imaginary number C. rational number D. whole number
3. What would be the hypothesis of the mathematical induction for x(x + 1) < x! , where x ≥ 7?
A. It is assumed that at x = k, k(k + 1)! < k!
B. It is assumed that at x = k, k(k + 1)! > k!
C. It is assumed that at x = k, k(k + 1)! < (k + 1)!
D. It is assumed that at x = k, k(k + 1)(k + 2)! < k!
4. For any positive integer x, ________ is divisible by 5 (2 pts)
A. 5x2 + 5 B. 2x + 4 C. x4 + 5x D. 3x2 + 2
Type of relations for aRb if and only if a<= b+1
Solve the following. (10 pts each)
1. Prove P(n) = n2(n + 1)
2. Recurrence relation an = 2n with the initial term a1= 2.
From around 1 pm to 5 pm, you attended the party of your friend. Because you are close to each other, she let you get her gold necklace in the jewelry box on the table in her room. You agreed to your friend’s request to get the necklace in her room. As you enter the room and open the jewelry box, you see nothing. Your friend calls the police to accuse you that you steal her gold necklace. How can you prove that you are innocent? Give your statements.
Find the edge chromatic
numbers to colour the edges of the complete graph with four and five vertices
