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If the statement q ∧ r is true, determine all combinations of truth values for p and s such that the statement (q → [¬p ∨ s]) ∧ [¬s → r] is true.
f p is plotted versus a range of parameter value x the resulting defines ...a?
Let A = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4}
and define a relation R on A as follows:
For all x, y A, x R y ⇔ 3|(x − y).
It is a fact that R is an equivalence relation on A. Use set-roster notation to write the equivalence classes of R.
[0]=
[1]=
[2]=
[3]=
How many distinct equivalence classes does R have?
List the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets.)
MATHEMATICAL INDUCTION AND RECURRENC
Solve the following. (10 pts each)
1. Prove P(n) = n2 (n + 1)
2. Recurrence relation an = 2n with the initial term a1 = 2.
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
Solve the following. (mod4)
1. How many 3-digit number can be formed from digits 1 – 5 if:
a. If repetition is allowed?
b. If repetition is not allowed?
2. How many strings of length 4 can be formed using the letters
COMPUTER if repetitions are not allowed?
3. Out of 17 regions in the country, ten will be chosen to be included
in a survey. How many ways of selecting 10 out of 17 regions?
4. Expand the (2𝑥 − 𝑦)3 using binomial coefficient.
5. Find for the coefficient of the following:
a. x4y3 after the expansion of (x – 2y )7
b. x6y6 after the expansion of (2x + y )12
6. Find the 9th term of the expression (2x + y )12
COMBINATIONS (mod4)
1. Patrick has assignments in 5 subjects. He can only do two assignments. In how many ways can he do two assignments?
2. In how many ways can a group of 5 men and 3 women be made out of a total of 10 men and 6 women?
3. A box contains 6 red, 5 blue and 3 white balls. In how many ways can we select 3 balls such that
a. They are of different colors?
b. They are all red?
c. Two are blue and one is white?
d. Exactly 2 are blue?
e. None is white?
f. At least two are white?
BINOMIAL COEFFICIENTS
1. Expand the (2𝑚 − 2𝑏)3 using binomial coefficient.
2. Find for the coefficient of a5b5 ; (a - 4b )10
3. Find the 5th term after expanding the expression (3x – 4y)15
PIGEONHOLE PRINCIPLE
Show that in a group of 27 English words, there must be at least two that begin with the same letter.
PERMUTATIONS (mod4)
1. There are 6 people to be arranged in a line for a concert. How many arrangements are possible?
2. How many strings of length 5 can be formed using the letters QUALITY if
a. Repetitions are not allowed?
b. Repetitions are allowed?
c. Starts with letter L and repetition is not allowed?
3. A group of 25 people are going to run a race. The top three runners earn gold, silver, and bronze medals. How many arrangements are possible?
4. In how many different ways can the letters of the word "CHANGE" be arranged in such a way that the vowels always come together?
5. Find the number of permutations of the word INFORMATION.
COUNTING METHODS mod4:
1. How many 5-digit number can be formed from digits 0 – 6 if:
a. If repetition is allowed?
b. If repetition is not allowed?
c. If one (1) is not to be used as the 1st digit and repetition is not allowed?
d. If one (1) is not to be used as the 1st digit and repetition is allowed?
2. How many possible passwords are there for the following conditions: 3 digits followed by 2 letters followed by 4 digits?
Show that (p → q) ∧ (q → r) → (p → r) is a tautology
#339914 on Dec 2023