Lecture Notes on Discrete Mathematics doc

... even, respectively. Then prove that the functions f and g ◦ f are one-one but g is not one-one. Solution: By definiton, it is clear that f is indeed one-one and g is not one-one. But g ◦f(x) = g(f (x)) = g(2x) ... odd. Then prove that f is one- one. Is f onto? Solution: Let us use the contrapostive argument to prove that f is one-one. Let if possible f(x) = f(y), for some x, y ∈ N. Using the d...

Ngày tải lên: 31/03/2014, 12:20

... V d i B 2 Khuếch Đại Một Chiều 6/10 KTĐ & ĐT GVC. ThS. Nguyễn Như Phong 1999 ĐHBK CMRR _ ( Common Mode Rejection Ratio ) lg lg lg log ie E db m E ie fe E mE ie hR CMRR 20 20 g R h hR CMRR ... DA ). − Khuếch đại thuật toán ( Operational Amp_ opAmp ). T e R x R 1 R 3 R 2 Khuếch Đại Một Chiều 3/10 KTĐ & ĐT GVC. ThS. Nguyễn Như Phong 1999 ĐHBK KHUẾCH ĐẠI VI SA...

Ngày tải lên: 25/10/2013, 13:15

... chaotic. Continued fractions for rational numbers are always finite. Infinite periodic continued fractions correspond to “quadratic irrationalities”, that is, to roots of quadratic equations with rational ... 0. Proposition 1.3. Any rational number has a unique presentation as a finite continued fraction. Proof of existence. For an irreducible fraction p q , we shall prove the existence of a con...

Ngày tải lên: 08/03/2014, 12:20

... of Polynomials 93 Solutions to Testing Questions 101 Index 169 Lecture 1 Operations on Rational Numbers 1. Basic Rules on Addition, Subtraction, Multiplication, Division Commutative Law: a + ... element a belongs to the set A Contents Pr eface v Acknowledgments vii Abbreviations and Notations ix 1 Operations on Rational Numbers 1 2 Monomials and Polynomials 7 3 Linear Equations of Sin...

Ngày tải lên: 11/03/2014, 16:55

... derivative of the spin connection, since it’s not a tensor. So far our equations have been true for general connections; let’s see what we get for the Christoffel connection. The torsion-free requirement ... prop- erties of tensors. First consider the operation of contraction, which turns a (k, l) tensor into a (k −1, l −1) tensor. Contraction proceeds by summing over one upper and one lower i...

Ngày tải lên: 17/03/2014, 14:28

Ngày tải lên: 17/03/2014, 14:41

Ngày tải lên: 18/03/2014, 00:20

... simplification of the above conditions essentially leads to the definition of a one-way function. 2.2 One-Way Functions: Definitions In this section, we present several definitions of one-way functions. ... functions exist if and only if strong one way functions exist. Proof Sketch: By definition, a strong one way function is a weak one way function. Now assume that f is a weak one way function su...

Ngày tải lên: 28/03/2014, 20:20

... brought into the city of Buenos Aires each day for consumption, and no less than two tons of butter, one ton of cream, and three tons of cheese are used there daily. Argentina also exports ... alfalfa contains as much nutrition as two tons of wheat. The cost of growing alfalfa greatly depends upon the situation of the land to be dealt with; also upon whether labour is plentiful or ......

Ngày tải lên: 29/03/2014, 02:21

... people into a condition to construct such monuments, and when we reflect on the interval that must have passed after their construction until the epoch of their abandonment, we are constrained to ... than one or two skeletons of that people have been recovered in a condition suitable for intelligent examination. It is stated in the work of Squier and Davis that the only skull belonging inco...

Ngày tải lên: 30/03/2014, 23:20