Functions Definition of Function Terminology Addition and
... Let f be a function from A to B. Let S be a subset of B. Show that f-1(S) = f-1(S) Proof: We must show that f-1(S) ⊆ f-1(S) and that f-1(S) ⊆ f-1(S) . Let x ∈ f-1(S). Then x∈A and f(x) ∉ S. Since f(x) ∉ S, x ∉ f-1(S). Therefore x ∈ f-1(S). Now let x ∈ f-1(S). Then x ∉ f-1(S) which implies that f(x) ...
... Let f be a function from A to B. Let S be a subset of B. Show that f-1(S) = f-1(S) Proof: We must show that f-1(S) ⊆ f-1(S) and that f-1(S) ⊆ f-1(S) . Let x ∈ f-1(S). Then x∈A and f(x) ∉ S. Since f(x) ∉ S, x ∉ f-1(S). Therefore x ∈ f-1(S). Now let x ∈ f-1(S). Then x ∉ f-1(S) which implies that f(x) ...
Real Numbers and the Pythagorean Theorem
... 28. LOGIC Each statement below is true for square roots. Determine whether the statement is also true for cube roots. Explain your reasoning and give an example to support your explanation. a. You cannot find the square root of a negative number. b. Every positive number has a positive square root a ...
... 28. LOGIC Each statement below is true for square roots. Determine whether the statement is also true for cube roots. Explain your reasoning and give an example to support your explanation. a. You cannot find the square root of a negative number. b. Every positive number has a positive square root a ...
The Limit of a Sequence of Numbers
... We give next what is probably the most useful fundamental result about sequences, the BolzanoWeierstrass Theorem. It is this theorem that will enable us to derive many of the important properties of continuity, dierentiability, and integrability. Theorem 2: Bolzano-Weierstrass Every bounded sequenc ...
... We give next what is probably the most useful fundamental result about sequences, the BolzanoWeierstrass Theorem. It is this theorem that will enable us to derive many of the important properties of continuity, dierentiability, and integrability. Theorem 2: Bolzano-Weierstrass Every bounded sequenc ...
Summary of changes in Math 64 effective Fall 2014:
... Full coverage Solve quadratic equations using the quadratic formula (Emphasize) Use the discriminant to determine the number and type of solutions Determine the most efficient method to use when solving a quadratic equation Write quadratic equations from solutions Use the quadratic formula ...
... Full coverage Solve quadratic equations using the quadratic formula (Emphasize) Use the discriminant to determine the number and type of solutions Determine the most efficient method to use when solving a quadratic equation Write quadratic equations from solutions Use the quadratic formula ...
![Elliptic curves with Q( E[3]) = Q( ζ3)](http://s1.studyres.com/store/data/012817289_1-48789e28be1e09d726c6f57e08e36eee-300x300.png)
Elliptic curves with Q( E[3]) = Q( ζ3)
... odd prime powers q and for q|4 (see [1], Chap IX, Thm. I). On the other hand, there are counterexamples for q = 2t , t ≥ 3. The most famous of them was discovered by Trost (see [16]) and it is the diophantine equation x8 = 16, that has a solution in Qp , for all primes p ∈ Q, different from 2, but h ...
... odd prime powers q and for q|4 (see [1], Chap IX, Thm. I). On the other hand, there are counterexamples for q = 2t , t ≥ 3. The most famous of them was discovered by Trost (see [16]) and it is the diophantine equation x8 = 16, that has a solution in Qp , for all primes p ∈ Q, different from 2, but h ...
Coding Theory - Hatice Boylan
... It is not hard to show (see below) that limn→∞ Pn = 0. Therefore, our repetition code can improve a bad transmission to a one as good as we want, provided the transmission error p for bits is strictly less than 12 . What makes the repetition code so efficient is the fact that its two code words are ...
... It is not hard to show (see below) that limn→∞ Pn = 0. Therefore, our repetition code can improve a bad transmission to a one as good as we want, provided the transmission error p for bits is strictly less than 12 . What makes the repetition code so efficient is the fact that its two code words are ...
