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- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/08%3A_New_Page/8.03%3A_New_PageDefine addition by \[[\langle a, b\rangle]+[\langle c, d\rangle]:=[\langle a d+b c, b d\rangle]\] and multiplication by \[[\langle a, b\rangle] \cdot[\langle c, d\rangle]:=[\langle a \cdot c, b \cdot ...Define addition by \[[\langle a, b\rangle]+[\langle c, d\rangle]:=[\langle a d+b c, b d\rangle]\] and multiplication by \[[\langle a, b\rangle] \cdot[\langle c, d\rangle]:=[\langle a \cdot c, b \cdot d\rangle] .\] We define the linear ordering on \(\mathbb{Q}\) by \[[\langle a, b\rangle] \leq[\langle c, d\rangle] \quad \text { iff } \quad a \cdot d \leq b \cdot c .\] Through the construction of the rational numbers, we have used set operations to build mathematical structures with which you are…
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)This book is written for students who have taken calculus and want to learn what “real mathematics" is. We hope you will find the material engaging and interesting, and that you will be encouraged to ...This book is written for students who have taken calculus and want to learn what “real mathematics" is. We hope you will find the material engaging and interesting, and that you will be encouraged to learn more advanced mathematics. This is the second edition of our text. It is intended for students who have taken a calculus course, and are interested in learning what higher mathematics is all about. It can be used as a textbook for an "Introduction to Proofs" course, or for self-study.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/04%3A_New_Page/4.01%3A_New_PageIn this chapter we discuss the principle of mathematical induction. Be aware that the word induction has a different meaning in mathematics than in the rest of science. The principle of mathematical i...In this chapter we discuss the principle of mathematical induction. Be aware that the word induction has a different meaning in mathematics than in the rest of science. The principle of mathematical induction depends on the order structure of the natural numbers, and gives us a powerful technique for proving universal mathematical claims.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/07%3A_New_Page
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/zz%3A_Back_Matter/20%3A_GlossaryExample and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pag...Example and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages] (Optional) Caption for Image (Optional) External or Internal Link (Optional) Source for Definition "Genetic, Hereditary, DNA ...") (Eg. "Relating to genes or heredity") The infamous double helix CC-BY-SA; Delmar Larsen Glossary Entries Definition Image Sample Word 1 Sample Definition 1
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/06%3A_New_Page/6.05%3A_New_PageIf \(F\) is an instruction or rule (and hence a finite sequence of symbols from \(X\) ), then there is \(N \in \mathbb{N}\) such that \[F \in X^{N} \text {. }\] So it is easily seen that the set of al...If \(F\) is an instruction or rule (and hence a finite sequence of symbols from \(X\) ), then there is \(N \in \mathbb{N}\) such that \[F \in X^{N} \text {. }\] So it is easily seen that the set of all possible instructions for elements of \(\mathbb{N}^{\mathbb{N}}, I\), satisfies \[I \preceq \bigcup_{N \in \mathbb{N}} X^{N} .\] For \(N \in \mathbb{N}, X^{N}\) is the direct product of \(N\) factors of \(X\), and by Theorem \(6.20\), \[\left|X^{N}\right| \leq \aleph_{0} .\] The set \(\bigcup_{N …
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/07%3A_New_Page/7.04%3A_New_PageLet \(1 \leq n<k \leq p\) be such that \[a^{n} \equiv a^{k} \quad \bmod p .\] Then \[p \mid a^{k}-a^{n} .\] Hence \[p \mid a^{n}\left(a^{k-n}-1\right) .\] However \(p \nmid a^{n}\) and thus by Proposi...Let \(1 \leq n<k \leq p\) be such that \[a^{n} \equiv a^{k} \quad \bmod p .\] Then \[p \mid a^{k}-a^{n} .\] Hence \[p \mid a^{n}\left(a^{k-n}-1\right) .\] However \(p \nmid a^{n}\) and thus by Proposition 7.3, \[p \mid a^{k-n}-1 .\] Thus \[a^{k-n} \equiv 1 \quad \bmod p .\] Therefore \[o_{p}(a) \leq k-n<p .\] Proposition 7.16.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/09%3A_New_Page/9.05%3A_New_PageThen \(p\) can be factored into a product of linear factors \(\left(x-c_{k}\right)\) and quadratic factors \(\left(\left(x-a_{k}\right)^{2}+b_{k}^{2}\right)\) : \[p(x)=c\left(\prod_{k=1}^{N_{1}}\left(...Then \(p\) can be factored into a product of linear factors \(\left(x-c_{k}\right)\) and quadratic factors \(\left(\left(x-a_{k}\right)^{2}+b_{k}^{2}\right)\) : \[p(x)=c\left(\prod_{k=1}^{N_{1}}\left(x-c_{k}\right)\right)\left(\prod_{j=1}^{N_{2}}\left(\left(x-a_{j}\right)^{2}+b_{j}^{2}\right)\right)\] for some (not necessarily distinct) real numbers \(c, c_{j}, a_{j}, b_{j}\).
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/06%3A_New_Page/6.03%3A_New_PageWe enumerate all the elements in the range of \(f\); each one is a sequence of 0 ’s and 1 ’s. \[\begin{aligned} & f(0)=a_{00} \quad a_{01} \quad a_{02} \quad a_{03} \quad \ldots \quad a_{0 j} \ldots \...We enumerate all the elements in the range of \(f\); each one is a sequence of 0 ’s and 1 ’s. \[\begin{aligned} & f(0)=a_{00} \quad a_{01} \quad a_{02} \quad a_{03} \quad \ldots \quad a_{0 j} \ldots \\ & f(1)=a_{10} \quad a_{11} \quad a_{12} \quad a_{13} \quad \ldots \quad a_{1 j} \quad \ldots \\ & f(2)=a_{20} \quad a_{21} \quad a_{22} \quad a_{23} \quad \ldots \quad a_{2 j} \quad \ldots \\ & f(3)=a_{30} \quad a_{31} \quad a_{32} \quad a_{33} \quad \ldots \quad a_{3 j} \quad \ldots \end{aligned…
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/05%3A_New_Page/5.04%3A_New_PageLet \(a \in \mathbb{R}\) and suppose that \[\lim _{x \rightarrow a} g(x)=L_{1}\] and \[\lim _{x \rightarrow L_{1}} f(x)=L_{2} .\] Prove that \[\lim _{x \rightarrow a} f \circ g=L_{2} .\] If \(g\) is c...Let \(a \in \mathbb{R}\) and suppose that \[\lim _{x \rightarrow a} g(x)=L_{1}\] and \[\lim _{x \rightarrow L_{1}} f(x)=L_{2} .\] Prove that \[\lim _{x \rightarrow a} f \circ g=L_{2} .\] If \(g\) is continuous at \(a\) and \(f\) is continuous at \(g(a)\), is \(f \circ g\) continuous at \(a\) ?
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/09%3A_New_Page/9.07%3A_New_PageWhat are the primitive fourth roots of unity? Show that if \(\omega\) is any \(n^{\text {th }}\) root of unity other than 1, then \(1+\omega+\omega^{2}+\cdots+\omega^{n-1}=0 .\) EXERCISE 9.3. Let \(g:...What are the primitive fourth roots of unity? Show that if \(\omega\) is any \(n^{\text {th }}\) root of unity other than 1, then \(1+\omega+\omega^{2}+\cdots+\omega^{n-1}=0 .\) EXERCISE 9.3. Let \(g: G \rightarrow \mathbb{C}\) be a continuous function on \(G \subseteq \mathbb{C}\). Conversely, show that the continuity of \(\Re(g)\) and \(\Im(g)\) imply the continuity of \(g\). Show that every continuous real-valued function on a closed, bounded subset of \(\mathbb{C}\) attains its extrema.
