Note
... Definition: A function f is called an algebraic function if it is constructed by applying algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) to the polynomials. Examples: ...
... Definition: A function f is called an algebraic function if it is constructed by applying algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) to the polynomials. Examples: ...
Rational Numbers
... emphasize the analogy. The construction of the integers involved introducing ‘additive inverses’ for the natural numbers, and now the construction of the rational numbers involves the introduction of ‘multiplicative inverses’ for the non-zero integers. We must also introduce other new objects; besid ...
... emphasize the analogy. The construction of the integers involved introducing ‘additive inverses’ for the natural numbers, and now the construction of the rational numbers involves the introduction of ‘multiplicative inverses’ for the non-zero integers. We must also introduce other new objects; besid ...
Chapter 6
... • 2. Eliminate the denominators of the rational expressions by multiplying both sides of the equation by the LCD. • 3. Solve the resulting equation • 4. Check all solutions in original equation being careful of extraneous ...
... • 2. Eliminate the denominators of the rational expressions by multiplying both sides of the equation by the LCD. • 3. Solve the resulting equation • 4. Check all solutions in original equation being careful of extraneous ...
The Riemann Hypothesis for Elliptic Curves
... and extended analytically to the whole complex plane by a functional equation (see [8, p. 14]). The original Riemann hypothesis asserts that the nonreal zeros of the Riemann zeta function ζ(s) all lie on the line Re(s) = 1/2. In his monumental paper [11] of 1859, Riemann made this assertion in order ...
... and extended analytically to the whole complex plane by a functional equation (see [8, p. 14]). The original Riemann hypothesis asserts that the nonreal zeros of the Riemann zeta function ζ(s) all lie on the line Re(s) = 1/2. In his monumental paper [11] of 1859, Riemann made this assertion in order ...
Formal Methods Key to Homework Assignment 2, Part 3
... 67. Let A and B be integers. Prove that if AB is odd, then A + B is even. To prove this, we first prove a couple of familiar results. – AB is odd iff both A and B are odd. We need to see that AB odd implies both A and B are odd, and both A and B odd implies AB odd. First we give a direct proof that ...
... 67. Let A and B be integers. Prove that if AB is odd, then A + B is even. To prove this, we first prove a couple of familiar results. – AB is odd iff both A and B are odd. We need to see that AB odd implies both A and B are odd, and both A and B odd implies AB odd. First we give a direct proof that ...
Filters and Ultrafilters
... that i0 ∈ J is an ultrafilter. Clearly Ji0 is closed under oversets and finite intersections, so its a filter. It doesn’t contain the emptyset, so it is a proper filter. Suppose Ji0 is properly contained in some other filter J 0. Then there would be an element J 0 ∈ J 0 such that i0 6∈ J 0. However, ...
... that i0 ∈ J is an ultrafilter. Clearly Ji0 is closed under oversets and finite intersections, so its a filter. It doesn’t contain the emptyset, so it is a proper filter. Suppose Ji0 is properly contained in some other filter J 0. Then there would be an element J 0 ∈ J 0 such that i0 6∈ J 0. However, ...
