Theorem 2.10: When n + 1 pigeons roost in n holes, there must be some hole...
Theorem 2.10: When n + 1 pigeons roost in n holes, there must be some hole containing at least two pigeons.
Prove the Pigeonhole Principle using mathematical induction.
Homework Answers
Step 1:IF n=1 then there are two pigeons and one hole,so both pigeons roost in one hole.
Step 2:Lets take n=2 then there are 3 pigeons and 2 holes , so one pigeon roost in one hole and other 2 roost in one hole.
Step 3:taking some x , assume that if x+1 pigeons roost in x holes one hole will atleast contain 2 pigeons.
STEP 4:consider the case of x+2 pigeons among x+1 holes,choose one of the holes. It will atleast contains zero pigeons,one pigeon or two or more pigeon.IF it has two or more pigeons we proved it.If it has one pigeon then remaining x+1 pigeons roost in x holes.so by assumption one of these holes contain atleast 2 pigeons.IF it has zero pigeons then,remaining x+1 of the x+2 pigeons roost among x holes so as previously one of these x holes contains atleast two pigeons.
Hence Proved When n + 1 pigeons roost in n holes, there must be some hole containing at least two pigeons.
Add Answer to:
Theorem 2.10: When n + 1 pigeons roost in n holes, there must be
some hole...
Proposition PHP2. (The Pigeonhole Principle.) If n or more pigeons are distributed among k 0 pige...
Proposition PHP2. (The Pigeonhole Principle.) If n or more pigeons are distributed among k 0 pigeonholes, then at least one pigeonhole contains at least 1 pigeons. Proof. Suppose each pigeonhole contains at most 1-1 pigeons. Then, the total number of pigeons is at most k(P1-1) < k㈜ = n pigeons (because R1-1( , RI). Exercises. Prove: (a) If n objects are distributed among k>0 boxes, then at least one box contains at most L objects (b) Given t > 0...
a be a real number . If a--a, prove that either a 0 or a 1....
a be a real number . If a--a, prove that either a 0 or a 1. 8. (Pigeonhole Principle) Suppose we place m pigeons in n pigeonholes, where m and n are positive integers. If m > n, show that at least two pigeons must be placed in the same pigeonhole. [Hint (from Robert Lindahl of Morehead State University): For i 1, 2, . . . , n, let Xi denote the number of pigeons that are placed in the...
1. Use Pigeon hole principle to prove that any graph with at least 2 vertices contains...
1. Use Pigeon hole principle to prove that any graph with at
least 2 vertices contains two vertices of the same degree. (Hint:
Prove by contradiction. (4 points)
2. Given (6 Points)
a. Prove the above equation using binomial theorem. (3
Points)
b. Give a combinatorial proof for the given equating. (3
Points)
4n = (0)2" + (1)2" +...+)2"-
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE r=1(no induction required, just use the definition of the determ...
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE
r=1(no induction required, just use the definition of the
determinants)
Theorem 4.3. The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 Sr S n, we have ar-1 ar-1 ar-1 ar+1 ar+1 ar+1 an an rt whenever k is a scalar and u, v, and each a are row vectors in F". Proof. The proof...
Only need 2-5. Need it done ASAP, thank you in advance!! Proofs 1) (1.7.16) Prove that...
Only need 2-5. Need it done ASAP, thank you in advance!!
Proofs 1) (1.7.16) Prove that if m and n are integers and nm is even, then m is even or n is even. * What is the best approach here, direct proof, proof by contraposition, or proof by contradiction why? * Complete the proof. 2) Prove that for any integer n, n is divisible by 3 iff n2 is divisible by 3. Does your proof work for divisibility by...
1. Prove by induction that, for every natural number n, either 1 = n or 1<n....
1. Prove by induction that, for every natural number n, either 1 = n or 1<n. 2. Prove the validity of the following form of the principle of mathematical in duction, resting your argument on the form enunciated in the text. Let B(n) denote a proposition associated with the integer n. Suppose B(n) is known (or can be shown) to be true when n = no, and suppose the truth of B(n + 1) can be deduced if the truth...
Let P(n) be some propositional function. In order to prove P(n) is true for all positive...
Let P(n) be some propositional function. In order to prove P(n) is true for all positive integers, n, using mathematical induction, which of the following must be proven? OP(K), where k is an arbitrary integer with k >= 1 If P(k) is true, then P(k+1) is true, where k is an arbitrary integer with k >= 1 P(O) P(k+1), where k is an arbitrary integer with k>= 1
: Let a1, a2, a3, . . . be the sequence of integers defined by a1 = 1 and defined for n ≥ 2 by the recurrence relation a...
: Let a1, a2, a3, . . . be the sequence of integers defined by a1 = 1 and defined for n ≥ 2 by the recurrence relation an = 3an−1 + 1. Using the Principle of Mathematical Induction, prove for all integers n ≥ 1 that an = (3 n − 1) /2 .
In the following problem, we will work through a proof of an important theorem of arithmetic. Your job will be to read the proof carefully and answer some questions about the argument. Theorem (The Di...
In the following problem, we will work through a proof of an
important theorem of arithmetic. Your job will be to read the proof
carefully and answer some questions about the argument. Theorem
(The Division Algorithm). For any integer n ≥ 0, and for any
positive integer m, there exist integers d and r such that n = dm +
r and 0 ≤ r < m. Proof: (By strong induction on the variable n.)
Let m be an arbitrary...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may eq...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
Proposition PHP2. (The Pigeonhole Principle.) If n or more pigeons are distributed among k 0 pigeonholes, then at least one pigeonhole contains at least 1 pigeons. Proof. Suppose each pigeonhole contains at most 1-1 pigeons. Then, the total number of pigeons is at most k(P1-1) < k㈜ = n pigeons (because R1-1( , RI). Exercises. Prove: (a) If n objects are distributed among k>0 boxes, then at least one box contains at most L objects (b) Given t > 0...
a be a real number . If a--a, prove that either a 0 or a 1. 8. (Pigeonhole Principle) Suppose we place m pigeons in n pigeonholes, where m and n are positive integers. If m > n, show that at least two pigeons must be placed in the same pigeonhole. [Hint (from Robert Lindahl of Morehead State University): For i 1, 2, . . . , n, let Xi denote the number of pigeons that are placed in the...
1. Use Pigeon hole principle to prove that any graph with at
least 2 vertices contains two vertices of the same degree. (Hint:
Prove by contradiction. (4 points)
2. Given (6 Points)
a. Prove the above equation using binomial theorem. (3
Points)
b. Give a combinatorial proof for the given equating. (3
Points)
4n = (0)2" + (1)2" +...+)2"-
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE
r=1(no induction required, just use the definition of the
determinants)
Theorem 4.3. The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 Sr S n, we have ar-1 ar-1 ar-1 ar+1 ar+1 ar+1 an an rt whenever k is a scalar and u, v, and each a are row vectors in F". Proof. The proof...
Only need 2-5. Need it done ASAP, thank you in advance!!
Proofs 1) (1.7.16) Prove that if m and n are integers and nm is even, then m is even or n is even. * What is the best approach here, direct proof, proof by contraposition, or proof by contradiction why? * Complete the proof. 2) Prove that for any integer n, n is divisible by 3 iff n2 is divisible by 3. Does your proof work for divisibility by...
1. Prove by induction that, for every natural number n, either 1 = n or 1<n. 2. Prove the validity of the following form of the principle of mathematical in duction, resting your argument on the form enunciated in the text. Let B(n) denote a proposition associated with the integer n. Suppose B(n) is known (or can be shown) to be true when n = no, and suppose the truth of B(n + 1) can be deduced if the truth...
Let P(n) be some propositional function. In order to prove P(n) is true for all positive integers, n, using mathematical induction, which of the following must be proven? OP(K), where k is an arbitrary integer with k >= 1 If P(k) is true, then P(k+1) is true, where k is an arbitrary integer with k >= 1 P(O) P(k+1), where k is an arbitrary integer with k>= 1
In the following problem, we will work through a proof of an
important theorem of arithmetic. Your job will be to read the proof
carefully and answer some questions about the argument. Theorem
(The Division Algorithm). For any integer n ≥ 0, and for any
positive integer m, there exist integers d and r such that n = dm +
r and 0 ≤ r < m. Proof: (By strong induction on the variable n.)
Let m be an arbitrary...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
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