Prove that the following statements are false: a. The sum of two odd counting numbers is always even. 6. b. For all number x, x+5|=|x|+ 5.

Q: Which of the following statements best justifies In(n+ 1)

A: We are given the following series.               We know that ln(x) < x. So, ln(n+1)/n will…

Q: 2. Rewrite each of the following statements so that it is clear that each is an implication a. n+1…

A: This problem involves basic logic of number theory.

Q: Recall that P(n, r) is the number of r-permutations of an n-element set. Prove the following…

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Q: Prove the following by contrapositive: If x is an even integer, then x² is even.

A: Let x is not even, it has to prove that x is not an integer or x2 is not even. Let x is not even…

Q: 3. Let n and m be positive integers. Evaluate > (km+1 (k – 1)m+1) if possible. If not possible, k=1…

A: We have to evaluate the given summation.

Q: If X~N(μ , 5), Find μ if P(X>12) = 0.02118 1.85 22.15 2.03 11.89

A: Given information X ~ N(μ, 5) P(X > 12) = 0.02118 P(X<12) = 1 – 0.02118 = 0.97882 Z at p=…

Q: If x is an odd integer, then x² +3x+5 is odd.

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Q: Prove the following statement by contradiction. If x is odd, then 5x is odd

A: The given problem is to prove the following statement by using contradiction method. Given…

Q: 1. |x+y|≥x+y 2. 3=5, or 7 is a prime number. 3. If x is a prime number, then x is odd.

A: 1) For first statement we can show that by modulus of a number   2) One of the substatement is true…

Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: Prove the following claims: (a) If n is an integer between 0 and 10, then n = a² +b² +c² +d², for…

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Q: 9. This question is a little more challenging. How many integers in the interval [1, 99 999] (i.e.,…

A: Given data:   The given sum of all digits is k=10. The number of digits that can be used is n=5.…

Q: Prove the following statements: (a) If two integers are both odd, then their sum is even. (b) If two…

A: Consider two odd integer

Q: Prove: The nth Fibonacci number Fn is even if and only if 3 | n.

A: The nth Fibonacci number Fn is even if and only if 3|n.

Q: Let A be the set of integers that are multiples of 3 between 1 and 15 inclusive and B be the set of…

A: Multiples of 3 between 1 and 15 (inclusive) are {3,6,9,12,15}. Set of even natural numbers up to 20…

Q: 18. Prove the following proposition: For all integers a and b, if 3 divides (a? + b?), then 3…

A: Suppose 3 divides a2+b2 for all integers a and b To prove 3 divides a and 3 divides b Let us prove…

Q: Show that the following sequence of functions is a complete orthonor- mal system in L?([0, 7 ]):

A: In this question, we have to prove the sequence of the function is a complete orthonormal system in…

Q: Prove by using a direct proof that if n is an odd number, then n2 is also odd.

A:

Q: 75. If f(7) = 13 and fis one-to-one, what is f(13)?

A: For a function domain is set of values for which the function is defined. The range of a function is…

Q: Prove that the Ving of ComPelX numbers C¢,+.)is fielol

A: Solution is given as below:

Q: find a counterexample to show that the following conjecture is false. Conjecture: For all counting…

A: Given that, n3+5n+66.

Q: In which of the following sets are there in- finitely many prime numbers? {n? – n+ 41 : n E \N} {бn…

A:

Q: If 90% of x is 8100 then x is

A: Given,         90% of x is 8100.

Q: 1. Consider the following rules on the set of natural numbers, N, 1) write down 5 ordered pairs of…

A: The 5 order pair of the natural number that is satisfy the relation “less than or equal to” as…

Q: Prove the following statement by smallest counterexample. For every integer n≥0, 24 divides (52n…

A:

Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

Q: 1) a.The set of all whole numbers not greater than 5. b.The set of all counting numbers less than or…

A: To write give set into listing method

Q: Prove that the set N of natural Numbers is not bounded in R. Show that inf S = 0 where S = { ne N} n

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Q: In the given question as follows, determine whether the statement is true or false. If it is false,…

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Q: Prove directly from the definitions that for all integers m and n, if m and n have the same parity,…

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Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: Suppose that the following fact is proven by exhaustion. Theorem: Every integer in the range from…

A: Composite number:   A positive integer is said to be composite number if it can be formed by…

Q: Show by contradiction that for any integer n: if n>0, then n+10 n+5

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Q: prove by contrapositive : for all n,m integers if nm is even then n is even or m is even

A:

Q: If n is an integer greater than 2, then for all integers m, n does not divide m or n + m # nm.

A: c Given n∈Zand n>2. We have to prove that for all m∈Z, n/m or n+m/nm. Suppose n/m and…

Q: 11. Establish the following relation between Fibonacci and Lucas numbers: L, = un + 2un-1 for all n…

A: Remark: We know that both the Fibonacci numbers and  Lucas numbers are obtained as the sum of its…

Q: * 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is…

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Q: Prove or disprove the following: For any integer x, if x + 5 is even, then x + 1 is even.

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Q: In which of the following sets are there in- finitely many prime numbers? {n² – 1 : nɛ \N} {n² – n+…

A:

Q: For any positive integer n and r = 1,2, ...,n, we have G) = (","') + C" - }). True O False

A: The expression nr can be presented in the factorial form as follows. nr=n!(n-r)!r! The expression…

Q: Let P be the set of integers that are multiples of 3 between 1 and 10 and Q be the set of positive…

A:

Q: disprove the following statement. The sum of an integer and its cube such that x + x° is

A: A statement is given as the sum of an integer and its cube such that x+x3 is even. The objective is…

Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: The Pigeonhole Principle:  If there are p+1 or more pigeons, need to distribute among p pigeonholes,…

Q: if m is even and n is odd demonstrate that m*n= is even

A:

Q: 4. Give the indirect proof of the statement "if n ie an inkger and n'tl is odd hen n is even.

A: We prove it by contradiction.

Q: Given p: n is an odd integer q: n is an even integer. Which of the following statement cannot be…

A: Given statements p: n is an odd integer q: n is an odd integer  The answer can be found by verifying…

Q: Let, x belongs to positive integers. A, B and C are the sets expressed with the following set buile…

A: Given: A={x|x is a prime number not exceeding 15} B={x|x is a positive odd integer not exceeding 15}…

Q: If S is the set of all positive integers greater that 19, is S finite or infinite?

A: