Comparing sizes of sets
... Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A to B iff |A| = |B|. Let’s do it. . . Take arbitrary finite sets A and B. LR: Assume f : A → B is bijective. Then f is inje ...
... Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A to B iff |A| = |B|. Let’s do it. . . Take arbitrary finite sets A and B. LR: Assume f : A → B is bijective. Then f is inje ...
41(2)
... The first assertion we shall disprove states that there are infinitely many pairs of positive coprime integers x, y such that 2\y, x2 + y2 E D, and ...
... The first assertion we shall disprove states that there are infinitely many pairs of positive coprime integers x, y such that 2\y, x2 + y2 E D, and ...
Lecture-13
... the recursive step - an expression of the computation or definition in terms of itself There are many recursive definitions in mathematics. Consider the factorial function: n! = n * (n-1) * (n -2) * … * 2 * 1 The same function can be defined recursively by giving a base case and a recursive st ...
... the recursive step - an expression of the computation or definition in terms of itself There are many recursive definitions in mathematics. Consider the factorial function: n! = n * (n-1) * (n -2) * … * 2 * 1 The same function can be defined recursively by giving a base case and a recursive st ...
Distribution of Summands in Generalized
... Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P ...
... Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P ...
25(4)
... that the second and fifth lines of the chain of equalities above are the same, by virtue of (2.4) and (2.5). Some interesting results for particular values of a and b may be found. For example, with a = 0, b = 2, we have, by (2,5) and (2.8), = 4 ^ ' 1 } (x) + 2«n(2) = 4(1 + x2)^'^ ...
... that the second and fifth lines of the chain of equalities above are the same, by virtue of (2.4) and (2.5). Some interesting results for particular values of a and b may be found. For example, with a = 0, b = 2, we have, by (2,5) and (2.8), = 4 ^ ' 1 } (x) + 2«n(2) = 4(1 + x2)^'^ ...
Nonnegative k-sums, fractional covers, and probability of small
... stronger result asserting that at least a constant proportion of the nk k-sums are nonnegative. More precisely, we prove the following statement. Theorem 1.4 There exists a positive function g(δ, k), such that for any fixed k and δ and all sufficiently large n, the following holds. For any set of n ...
... stronger result asserting that at least a constant proportion of the nk k-sums are nonnegative. More precisely, we prove the following statement. Theorem 1.4 There exists a positive function g(δ, k), such that for any fixed k and δ and all sufficiently large n, the following holds. For any set of n ...
