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  • 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.
  • 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!>2k 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!>2k and would need to prove that (k+1)!>2k+1. Hence, by the Second Principle of Mathematical Induction, we conclude that P(n) is true for all nN with n2, and this means that each natural number greater than 1 is either a prime number or is a product of prime numbers.
  • 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.
  • 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 ...T×BA×BA×(BC)=(A×B)(A×C)A×(BC)=(A×B)(A×C)A×(BC)=(A×B)(A×C)
  • 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.
  • https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/06%3A_Functions
  • 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 a2+b2=c2.
  • 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.
  • 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...Basic Properties(Ac)c=AAB=ABcEmpty Set, Universal Set             A=A and AU=c=U and Uc=De Morgan's Laws(AB)c=AcBc(AB)c=AcBcSubsets and ComplementsAB if and only if BcAc.
  • https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/06%3A_Functions/6.04%3A_Composition_of_Functions
    There are several ways to combine two existing functions to create a new function. For example, in calculus, we learned how to form the product and quotient of two functions and then how to use the pr...There are several ways to combine two existing functions to create a new function. For example, in calculus, we learned how to form the product and quotient of two functions and then how to use the product rule to determine the derivative of a product of two functions and the quotient rule to determine the derivative of the quotient of two functions.
  • https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.03%3A_Proof_by_Contradiction
    This is done by assuming that X is false and proving that this leads to a contradiction. (The contradiction often has the form R, where R is some statement.) When this h...This is done by assuming that X is false and proving that this leads to a contradiction. (The contradiction often has the form R \wedge \urcorner R, where R is some statement.) When this happens, we can conclude that the assumption that the statement X is false is incorrect and hence X cannot be false.

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