1. Which of the following represents all real numbers greater than -4 but not more than 8. A) [-4,8) B) [-4,8] C) (-4, 8) D) (-4,8]

Q: 2. Prove that " The sum of two odd numbers is even: using direct proof.

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Q: Suppose that a1, a2, a3, a4 are four numbers. 4 If > ak -9 then (3ak +5) is equal to.. k=1

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Q: Derive one-one correspondence between the real numbers in (0,1) and the set of irrational numbers in…

A: Both the sets that is real number in 0,1 and the set of irrational numbers in 0,1 are uncountable as…

Q: How many 5 digit even numbers are there, a. Given that the first digit cannot be zero. b. Given…

A: In five digit following digit can be filled in following ways: First digit can be filled by…

Q: fa,,a,,a,,...-,a, be an AP of non - zero terms, prove that ---- 1 +. 1 +...+ n-1 aa, a,a, a a, a,a

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Q: 3. Prove or disprove (x, y are real numbers a) [x- y] = [x] – by] TRUE FALSE because:

A: Introduction: The integer, that is largest and less than or equal to the number a is known as floor…

Q: What is the supremum of the set consisting the following real numbers: 0.1222...,…

A: Given Set of Numbers {0.1222....,0.2122...,0.22122....,0.222122...,........} Supremum of this set is…

Q: Two distinct numbers are selected from the set (1,2,3,4,...,36,37) so that the sum of the remaining…

A: Given two distinct numbers are selected from the set 1,2,3,4,…,36,37 and The sum of remaining 35…

Q: 5.The sum of 100 real numbers in a given set X equals zero. Prove that the sums of at least 99 pairs…

A: Probability is the branch of mathematics concerning numerical descriptions of how likely an event is…

Q: Find the smallest distinct postive numbers that provide a counterexample to show the statement is…

A: Let the odd numbers be 1 and 3 and the even number be 2, According to the question, The sum of any…

Q: . Since 10 is the "middle" value of {9,9,10,10,17}, the …………of H is 10.

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Q: Let A be the set of positive integers that are less than or equal to 40 and are multiples of 4 and…

A: Given: Let A be the set of positive integers that are less than or equal to 40 and are multiples of…

Q: How many different 11-letter words (real imaginary) can be formed from the letters in the word…

A: We have to find the total number of 11-letter words that can be formed from the letter in the word…

Q: Prove (kindly provide reason/s) that for any real numbers m and n, (-m ) + ( - n ) = - ( m + n)

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Q: How many ways are there to list the digits {1,2, 2, 3, 4, 5, 6} so that identical digits are not in…

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Q: 6. Show that Vz eC i) z-7 is a real number. ii) (z-2) is a real number.

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Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: 2, then the conju

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Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

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Q: 9. Prove that there are as many positive real numbers as there are real numbers.

A: Consider the set X and Y. Then we said that |X| = |Y| if there exists bijective function from X to…

Q: Prove that 3:8 is equivalent to 12:32

A: According to the question, we have to prove that the ratio 3:8 is equivalent to 12:32. A ratio is a…

Q: EIf a,,a,,a,.--,a, be an A P of non - zero terms, prove that ------ 1 1 +. 1 n-1 %3D a,a, a,a, a,a,

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Q: How many positive integers less than 10000 are there which contain at least one 2 or at least one 3…

A: The objective is to evaluate the number of positive integers less that 10000 that contain at least…

Q: Dots and dashes can be used to represent symbols (Ex: C = -.-. ,Q = --.-). How many symbols could we…

A: We have two characters: dots and dashes. We must represent every symbol using just dots and dashes.…

Q: The height of 10 plants is measured in inches and found to be 12,6,15,3,12,621,15,18 and 12. Verify…

A: We know that, Mean, X bar = (summation Xi)/m Here n= 10 X bar = (12+6+15+3+12+6+21+15+18+12)/10 = 12

Q: The set of numbers {0, 1, 2, 3, 4, ...} is known as the negative integers. whole numbers. positive…

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Q: The sum of all positive integers less than 300 which are not divisible by 3.

A: Sum of n positive integer is given by, Sn=nn+12

Q: Show that there are infinitely many examples of two consecutive positive integers that cannot be…

A: According to the given information, It is required to prove that there are infinitely many examples…

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: Let a, b and c be integers that satisfy 1 1 1 - - - - a b C |(A) 2022 (В) 2025 Find one possible…

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Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

A: Given an=1000nn!. We have to find the N∈I such that an+1<an, n>N. For n=1 a1=100011! For n=2…

Q: How many ways are there to list the digits 1, 1, 2, 2, 3, 4, 5 so that the digit 3 and 4 are not in…

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Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: 3. Show that (5+ 13)e + (5- i3)-" is a real number if is also a real number.

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Q: What are all of the natural numbers divisible by 8 whose sum of digits is less than 10 and the…

A: Given the requirements

Q: The sum of all positive integers less than 400 which are divisible by 5.

A: Positive integers less than 400 which are divisible by 5 are⇒n=79 5,10,15,20,...,395 The above…

Q: = there to construct a string of 4 digits if numbers can be repeated?

A: We have to make a 4-digit number where digits can be repeated. Since the digits of number are not…

Q: of the apples is equal to 3 of the oranges. If there are 12 mor 5 In a fruit stall, - - annles than…

A: Fraction:  A small part of something.  In other words, numerical value that is not a whole number A…

Q: Students are sometimes uncertain about which zeros in decimals can be dropped and which can't (with-…

A: We have to tell when can zeros in decimals are dropped and which can't without changing the meaning…

Q: Using only the Axioms and Elementary Properties of the real numbers, prove Cauchy's Inequality: for…

A: xy≤12x2+y2 For all real numbers x,y.

Q: sum of two even integers is even

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Q: Prove, that (nezond n73) If x"=5,+hen x is not a rational number.

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Q: What is the set of O of odd positive integers less than 10?

A: The odd positive integers less than 10 are 1, 3, 5, 7, 9.

Q: Suppose that a1, a2, a3, a4, a5 are five numbers. 5 If ak =-7 then (2ak + 3) is equal to... k=1 k=1

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Q: In justifying that 5- (-2) = 7, we could add a 0 to the 5 and replace it with: O 2-2 O -2-(-2). O…

A: In justifying that, 5 - (-2) = 7, we could add a 0 to the 5 and replace it with,

Q: Show that if seven integers are selected from the first 10 positive integers, there must be at least…

A: We can solve this question by pigeonhole principle

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Two sets A and B contains a and belements respectively. If power set of A contains 32 more elements…

A: If a set S contains n number of elements then the number of elements in its power set is 2n here…

Q: How many words of lengthn≥6 in the alphabet{1,2,3}contain at least two of each digit?(Prove a closed…

A: Here we have to find the total number of possible ways of length n≥6 in the alphabet {1,2,3} contain…