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3: Constructing and Writing Proofs in Mathematics
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/03%3A_Constructing_and_Writing_Proofs_in_Mathematics
A proof in mathematics is a convincing argument that some mathematical statement is true. A proof should contain enough mathematical detail to be convincing to the person(s) to whom the proof is addre...A proof in mathematics is a convincing argument that some mathematical statement is true. A proof should contain enough mathematical detail to be convincing to the person(s) to whom the proof is addressed. In essence, a proof is an argument that communicates a mathematical truth to another person (who has the appropriate mathematical background). A proof must use correct, logical reasoning and be based on previously established results.
4.2: Other Forms of Mathematical Induction
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/04%3A_Mathematical_Induction/4.02%3A_Other_Forms_of_Mathematical_Induction
Suppose that we want to prove that if $P(k)$ is true, then $P(k + 1)$ is true. (This could be the inductive step in an induction proof.) To do this, we would be assuming that $k! > 2^k$ and woul...Suppose that we want to prove that if $P(k)$ is true, then $P(k + 1)$ is true. (This could be the inductive step in an induction proof.) To do this, we would be assuming that $k! > 2^k$ and would need to prove that $(k + 1)! > 2^{k + 1}$ . Hence, by the Second Principle of Mathematical Induction, we conclude that $P(n)$ is true for all $n \in \mathbb{N}$ with $n \ge 2$ , and this means that each natural number greater than 1 is either a prime number or is a product of prime numbers.
7.1: Relations
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/07%3A_Equivalence_Relations/7.01%3A_Relations
The notion of a function can be thought of as one way of relating the elements of one set with those of another set (or the same set). A function is a special type of relation in the sense that each e...The notion of a function can be thought of as one way of relating the elements of one set with those of another set (or the same set). A function is a special type of relation in the sense that each element of the first set, the domain, is “related” to exactly one element of the second set, the codomain. This idea of relating the elements of one set to those of another set using ordered pairs is not restricted to functions.
Mathematical Reasoning - Writing and Proof (Sundstrom)
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)
Mathematical Reasoning: Writing and Proof is designed to be a text for the ﬁrst course in the college mathematics curriculum that introduces students to the processes of constructing and writing proof...Mathematical Reasoning: Writing and Proof is designed to be a text for the ﬁrst course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.
Appendix B: Answers for the Progress Checks
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/zz%3A_Back_Matter/22%3A_Appendix_B%3A_Answers_for_the_Progress_Checks
\(\begin{array} {lcl} {T \times B \subseteq A \times B} & & {A \times (B \cup C) = (A \times B) \cup (A \times C)} \\ {A \times (B \cap C) = (A \times B) \cap (A \times C)} & & {A \times (B - C) = (A ... $\begin{array} {lcl} {T \times B \subseteq A \times B} & & {A \times (B \cup C) = (A \times B) \cup (A \times C)} \\ {A \times (B \cap C) = (A \times B) \cap (A \times C)} & & {A \times (B - C) = (A \times B) - (A \times C)} \end{array}$
6.2: More about Functions
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/06%3A_Functions/6.02%3A_More_about_Functions
We have also seen various ways to represent functions. We have also seen that sometimes it is more convenient to give a verbal description of the rule for a function. In cases where the domain and cod...We have also seen various ways to represent functions. We have also seen that sometimes it is more convenient to give a verbal description of the rule for a function. In cases where the domain and codomain are small, finite sets, we used an arrow diagram to convey information about how inputs and outputs are associated without explicitly stating a rule. In this section, we will study some types of functions, some of which we may not have encountered in previous mathematics courses.
6: Functions
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/06%3A_Functions
1.2: Constructing Direct Proofs
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/01%3A_Introduction_to_Writing_Proofs_in_Mathematics/1.02%3A_Constructing_Direct_Proofs
The study of Pythagorean triples began with the development of the Pythagorean Theorem for right triangles, which states that if $a$ and $b$ are the lengths of the legs of a right triangle and \(c...The study of Pythagorean triples began with the development of the Pythagorean Theorem for right triangles, which states that if $a$ and $b$ are the lengths of the legs of a right triangle and $c$ is the length of the hypotenuse, then $a^2 + b^2 = c^2$ .
2.S: Logical Reasoning (Summary)
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/02%3A_Logical_Reasoning/2.S%3A__Logical_Reasoning_(Summary)
De Morgan's Laws $\urcorner (P \wedge Q) \equiv \urcorner P \vee \urcorner Q$ Biconditional Statement $(P \leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)$ Distributive Laws \(P \vee (Q \wedge...De Morgan's Laws $\urcorner (P \wedge Q) \equiv \urcorner P \vee \urcorner Q$ Biconditional Statement $(P \leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)$ Distributive Laws $P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)$ $P \wedge (Q \vee R) \equiv (P \wedge Q) \vee (P \wedge R)$ Conditionals with Disjunctions $P \to (Q \vee R) \equiv (P \wedge \urcorner Q) \to R$ $P \vee Q) \to R \equiv (P \to R) \wedge (Q \to R)$
8.3: Linear Diophantine Equations
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/08%3A_Topics_in_Number_Theory/8.03%3A_Linear_Diophantine_Equations
Very little is known about Diophantus’ life except that he probably was the first to use letters for unknown quantities in arithmetic problems. His famous work, Arithmetica, consists of approximately ...Very little is known about Diophantus’ life except that he probably was the first to use letters for unknown quantities in arithmetic problems. His famous work, Arithmetica, consists of approximately 130 problems and solutions; most of solutions of equations in various numbers of variables. While Diophantus did not restrict his solutions to the integers and recognized rational number solutions as well, today, however, the solutions for a so-called Diophantine equation must be integers.
5.S: Set Theory (Summary)
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/05%3A_Set_Theory/5.S%3A_Set_Theory_(Summary)
\[\begin{array} {ll} {\text{Basic Properties}} & & {(A^c)^c = A} \\ {} & & {A - B = A \cap B^c} \\ {\text{Empty Set, Universal Set}\ \ \ \ \ \ \ \ \ \ \ \ \ } & & {A - \emptyset = A \text{ and } A - U... $\begin{array} {ll} {\text{Basic Properties}} & & {(A^c)^c = A} \\ {} & & {A - B = A \cap B^c} \\ {\text{Empty Set, Universal Set}\ \ \ \ \ \ \ \ \ \ \ \ \ } & & {A - \emptyset = A \text{ and } A - U = \emptyset} \\ {} & & {\emptyset ^c = U \text{ and } U^c = \emptyset} \\ {\text{De Morgan's Laws}} & & {(A \cap B)^c = A^c \cup B^c} \\ {} & & {(A \cup B)^c = A^c \cap B^c} \\ {\text{Subsets and Complements}} & & {A \subseteq B \text{ if and only if } B^c \subseteq A^c.} \end{array}$

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