Let N= {1, 2, 3, 4, .} be the set of natural numbers and S= (1, 4, 9, 16, ...} be the set of squares of the natural numbers. Then N - S, since we have the one-to-one correspondence 1 + 1, 2 + 4, 3 + 9, 4 + 16, ... n+ n?. (This example is interesting, since it shows that an infinite set can be equivalent to a proper subset of itself.) Show that each of the following pairs of sets are equivalent by carefully describing a one-to-one correspondence between the sets. Complete parts (a) through (c) below. (a) The whole numbers and natural numbers, W = {0, 1, 2, 3, ..} and N= {1, 2, 3, 4, ...} Which of the following describes a one-to-one correspondence between the two sets? O A. For each element in W, there is an element in N that is double that element. O B. For each element in w. there is an element in N that is 1 areater than double that element.

Q: Write a proof that proves the Union Un an element of natural numbers (-1/n, 2-1/n)=(-1, 2). Also…

A: The given interval is (-1/n,2-1/n) where natural number n belongs to positive integers. So that we…

Q: Let J={1,2,3,…}, the set of counting numbers and let R be the set of real numbers. Is R-J countable?…

A:

Q: Let a,b,c,d be real number with a < b < c < d. Express the set [a,b] U [c,d] as the difference of…

A: We have given that, a, b, c, d are real numbers with a &lt; b &lt; c &lt; d. We need to express a, b…

Q: 4. The following table presents a sequence of algebraic expressions that have a logical sense. There…

A:

Q: Prove: The sum of an odd number of odd numbers must be odd.

A: Let us prove this by induction, that if n is odd and a1,a2,. . .. .an are odd, then ∑i=1nai is odd.…

Q: Use a number line to find each difference. 1) 7-8 ++++H 2) -5- (-9) ++++++ ++++

A:

Q: Let A be the set of odd integers, and let B be the set of positive integers less than 10. Find An B…

A:

Q: How many triples of integers from 2 through 16 can be formed in which the integers are in increasing…

A: Any triple of integers (i, j, k) with 2 ≤ i ≤ j ≤ k ≤ 16 can berepresented as a string of 15 – 1…

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

A:

Q: Show that given any set of 5 natural numbers, there are 3 numbers whose sum is divisible by 3.

A: Let q∈N. Note that, for every q there exists some x∈N and some v∈0,1,2 such that q=3x+v. Now, let…

Q: • Show that if m divides 3, then m divides 9. (We say a divides b if there exists an integer k such…

A: Let  m divides 3 .  Then there exists an integer k such that 3 = mk. Now, 9 can be written as…

Q: 3. Show that there exist 100 consecutive positive integers that are not perfect squares.

A: Here we have to show that there exists 100 consecutive positive integer that are not perfect square.…

Q: 20) Show that the sum of two consecutive integers is always odd.

A: Let x be any integer. Then its consecutive integer can be written as x+1.

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.