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Discrete Mathematics
Prove the following statement by contradiction.
“There is no integer solution to the equation
(x^2)-6=0 ."
Prove and disprove the following statements.
(a) For all the odd integers n, n^4 mod 8 = 1
(b) There exist positive integers a, b and ab, which all the three belong to perfect
squares.”
State the contrapositive for the statement given below and hence, provide a proof for the
statement.
“For all integers a and b, if ab is even, then at least one of a or b is even.”
Prove or disprove the following statement.
“Let p and q be positive integers, if p mod 4 = 1 and q mod 4 = 2, then pq mod 4 = 2.”
“There is no integer solution to the equation
(x^2)-6=0 ."
Prove and disprove the following statements.
(a) For all the odd integers n, n^4 mod 8 = 1
(b) There exist positive integers a, b and ab, which all the three belong to perfect
squares.”
State the contrapositive for the statement given below and hence, provide a proof for the
statement.
“For all integers a and b, if ab is even, then at least one of a or b is even.”
Prove or disprove the following statement.
“Let p and q be positive integers, if p mod 4 = 1 and q mod 4 = 2, then pq mod 4 = 2.”
Discrete Mathematics
Prove or disprove the following statement.
“Let p and q be positive integers, if p mod 4 = 1 and q mod 4 = 2, then pq mod 4 = 2.”
“Let p and q be positive integers, if p mod 4 = 1 and q mod 4 = 2, then pq mod 4 = 2.”
Discrete Mathematics
The boys’ and girls’ track teams have both won their regional tournament, and they decide to stay in the tournament city for an extra couple of hours to have a victory dinner. The coach of the boys’ team calls the parents of three student athletes to inform them of the delay. These parents each call three other households to tell them the news, and the pattern continues.
If Round 1 is the initial three phone calls made by the coach, how many phone calls will be made during Round 4?
If Round 1 is the initial three phone calls made by the coach, how many phone calls will be made during Round 4?
Discrete Mathematics
What are the elements of the set S={x∈(-100,-50)┤| x≡6 mod 17}?
Discrete Mathematics
Prove that for any integer n such that n≡2 mod 3 , n^k≡1 or 2 (mod 3) , where k∈N
Discrete Mathematics
Prove the following statement using the first principle of Mathematical Induction:
Σ
Σ
Discrete Mathematics
1. The number 3-540-97285-9 is obtained from a valid ISBN number by switching two
consecutive digits. Find the ISBN number. (You may write a code for this)
2. The number 0-31-030369-0 is obtained from a valid ISBN number by switching two
consecutive digits. Find the ISBN number. (The same code for question 14 should work for
this problem with a different input data too).
3. Prove by Principle of Mathematical Induction (PMI) that 3 | (4m3 + 5m) for every
nonnegative integer n
4. Prove by PMI that for every positive integer k,
1
2 *3
+
1
3* 4
+...........+
1
(k +1)*(k + 2)
=
k
2 * k + 4
5. Find a formula for 2+4+6+ ………+2m for every positive integer m and then verify your
formula by the PMI.
6. Let k ≥ 1 be an integer. Prove by PMI that 1+ k + k2 +..........+ kn =
kn+1 −1
k −1
consecutive digits. Find the ISBN number. (You may write a code for this)
2. The number 0-31-030369-0 is obtained from a valid ISBN number by switching two
consecutive digits. Find the ISBN number. (The same code for question 14 should work for
this problem with a different input data too).
3. Prove by Principle of Mathematical Induction (PMI) that 3 | (4m3 + 5m) for every
nonnegative integer n
4. Prove by PMI that for every positive integer k,
1
2 *3
+
1
3* 4
+...........+
1
(k +1)*(k + 2)
=
k
2 * k + 4
5. Find a formula for 2+4+6+ ………+2m for every positive integer m and then verify your
formula by the PMI.
6. Let k ≥ 1 be an integer. Prove by PMI that 1+ k + k2 +..........+ kn =
kn+1 −1
k −1
Discrete Mathematics
Is {1,1,1,10}={1,1,1,1,10}?Justify your answer
Discrete Mathematics
Consider the Birthday Problem discussed in lecture. In this problem we calculate the probability
that, in a group of n people, at least two have the same birthday.
Let E be the event that at least two people share a birthday. In order to calculate P(E), we first need a
sample space. A possible sample space consists of n-tuples of the integers 1 . . . 365 (each of n people
have a birthday on one of the 365 days of the year; leap years are not considered).
(a) List or otherwise describe the sample space for n = 200. What is the size of the sample space?
that, in a group of n people, at least two have the same birthday.
Let E be the event that at least two people share a birthday. In order to calculate P(E), we first need a
sample space. A possible sample space consists of n-tuples of the integers 1 . . . 365 (each of n people
have a birthday on one of the 365 days of the year; leap years are not considered).
(a) List or otherwise describe the sample space for n = 200. What is the size of the sample space?
Discrete Mathematics
Prove or disprove that:
i) A-B⊆A-C-B
ii)(A-B)∪(B-C)∪(C-A)=(A∪B∪C)-(A∩B∩C)
i) A-B⊆A-C-B
ii)(A-B)∪(B-C)∪(C-A)=(A∪B∪C)-(A∩B∩C)
