Search & Filtering
p → ~q
MATHEMATICAL INDUCTION AND RECURRENCE
5. If P(k) = k2 (k + 2)(k – 1) is true, then what is P (k + 1)? (2 pts)
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? (2 pts)
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
MATHEMATICAL INDUCTION AND RECURRENCE
Solve the following. (10 pts each)
1. Prove P(n) = n2 (n + 1)
2. Recurrence relation an = 2n with the initial term a1 = 2.
RULE OF INFERENCE. Determine if the following argument is valid. If it is valid, what rule of inference is
used in each of the following arguments? Show solution. (4 pts each)
1. Joy wrote a C++ source code, or Jen wrote a Java source code. If Joy wrote a C++ source code, then the
problem was solved. If Jen wrote a Java source code, then the problem was solved.
2. There does not exist someone who likes to be COVID – 19 positive; hence, everyone does not like to be
vaccinated.
C. RELATION. Consider X = {-3, -2, -1, 0, 1} defined by (x,y) ∈ R if x≥ y.
Find for:
1. Elements of R (3 pts)
2. Domain and Range of R (2 pts)
3. Draw the digraph (3 pts)
4. Identify the properties of R (2pts)
E. PREDICATE LOGIC. Rewrite each sentence symbolically and determine the truth values. Write T if it is
true and F if it is false. Show complete solution. (5 pts each)
1. For some integer x, 𝑥 = 𝑥2 − 2
2. For every real number x, 𝑖𝑓 𝑥2 − 1 > 𝑥 𝑡ℎ𝑒𝑛 𝑥 + 1 > 1
3. For some integer n, 4n = 3n + 1
F. RULE OF INFERENCE. Determine if the following argument is valid. If it is valid, what rule of inference is
used in each of the following arguments? Show solution. (4 pts each)
1. Joy wrote a C++ source code, or Jen wrote a Java source code. If Joy wrote a C++ source code, then the problem was solved. If Jen wrote a Java source code, then the problem was solved.
2. There does not exist someone who likes to be COVID – 19 positive; hence, everyone does not like to be vaccinated.
PROBLEM SOLVING.
A. SET. Let A, B and C are sets and U be universal set.
U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e}
A = {-1, 1, 2, 4}
B = {0, 2, 4, 6}
C = {b, c, d}
Find for the following. Show complete solutions. (3 pts each)
1. 𝐵 ∪ 𝐶
2. 𝐴 − 𝐵 𝑥 𝐶
3. 𝑃𝑜𝑤𝑒𝑟 𝑠𝑒𝑡 𝑜𝑓 𝐶
4. |𝑃(𝐵)|
B. SEQUENCES. Consider the sequence {Sn} defined by Sn = 2n – 5, where 𝒏 ≥ −𝟏.
Find for:
1. ∑1𝑖=−1 𝑆𝑖 (5 pts)
2. ∏ 𝑆𝑖 4𝑖=2 (5 pts)
C. RELATION. Consider X = {-3, -2, -1, 0, 1} defined by (x,y) ∈ R if x ≥ y.
Find for:
1. Elements of R (3 pts)
2. Domain and Range of R (2 pts)
3. Draw the digraph (3 pts)
4. Identify the properties of R (2pts)
. PREDICATE LOGIC. Rewrite each sentence symbolically and determine the truth values. Write T if it is
true and F if it is false. Show complete solution. (5 pts each)
1. For some integer x, 𝑥 = 𝑥2 − 2
2. For every real number x, 𝑖𝑓 𝑥2 − 1 > 𝑥 𝑡ℎ𝑒𝑛 𝑥 + 1 > 1
3. For some integer n, 4n = 3n + 1
F. RULE OF INFERENCE. Determine if the following argument is valid. If it is valid, what rule of inference is
used in each of the following arguments? Show solution. (4 pts each)
1. Joy wrote a C++ source code, or Jen wrote a Java source code. If Joy wrote a C++ source code, then the problem was solved. If Jen wrote a Java source code, then the problem was solved.
2. There does not exist someone who likes to be COVID – 19 positive; hence, everyone does not like to be vaccinated.
PROBLEM SOLVING.
A. SET. Let A, B and C are sets and U be universal set.
U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e}
A = {-1, 1, 2, 4}
B = {0, 2, 4, 6}
C = {b, c, d}
Find for the following. Show complete solutions.
1. 𝐵 ∪ 𝐶
2. 𝐴 − 𝐵 𝑥 𝐶
3. 𝑃𝑜𝑤𝑒𝑟 𝑠𝑒𝑡 𝑜𝑓 𝐶
4. |𝑃(𝐵)|
B. SEQUENCES. Consider the sequence {Sn} defined by Sn = 2n – 5, where 𝒏 ≥ −𝟏.
Find for:
1. ∑1𝑖=−1 𝑆𝑖 (5 pts)
C. RELATION. Consider X = {-3, -2, -1, 0, 1} defined by (x,y) ∈ R if x ≥ y.
Find for:
1. Elements of R
2. Domain and Range of R
3. Draw the digraph
4. Identify the properties of R
