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- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/07%3A_New_Page/7.04%3A_New_PageWe define the sum and product of congruence classes in \(\mathbb{Z}/n\mathbb{Z}\) via \[[a]_n + [b]_n:= [a+b]_n \quad \text{and} \quad [a]_n \cdot [b]_n:= [a\cdot b]_n.\] However, it is possible for \...We define the sum and product of congruence classes in \(\mathbb{Z}/n\mathbb{Z}\) via \[[a]_n + [b]_n:= [a+b]_n \quad \text{and} \quad [a]_n \cdot [b]_n:= [a\cdot b]_n.\] However, it is possible for \([a]_n\cdot[b]_n = [0]_n\) even when \([a]_n \neq [0]_n\) and \([b]_n \neq [0]_n\). If \(n\in \mathbb{N}\) such that \(n\) is not prime, then there exists \([a]_n, [b]_n \in \mathbb{Z}/n\mathbb{Z}\) such that \([a]_n\cdot[b]_n = [0]_n\) while \([a]_n \neq [0]_n\) and \([b]_n \neq [0]_n\).
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/03%3A_New_Page/3.02%3A_New_PageIn the 1920s, adjustments to Zermelo’s axioms were made by Abraham Fraenkel (1891–1965), Thoralf Skolem (1887–1963), and Zermelo that resulted in a collection of nine axioms, called ZFC, where ZF stan...In the 1920s, adjustments to Zermelo’s axioms were made by Abraham Fraenkel (1891–1965), Thoralf Skolem (1887–1963), and Zermelo that resulted in a collection of nine axioms, called ZFC, where ZF stands for Zermelo and Fraenkel and C stands for the Axiom of Choice, which is one of the nine axioms.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/04%3A_New_Page/4.01%3A_New_PageIn this chapter, we introduce mathematical induction, which is a proof technique that is useful for proving statements of the form (∀n ∈ \(\mathbb{N}\))P (n), or more generally (∀n ∈ \(\mathbb{Z}\))(n...In this chapter, we introduce mathematical induction, which is a proof technique that is useful for proving statements of the form (∀n ∈ \(\mathbb{N}\))P (n), or more generally (∀n ∈ \(\mathbb{Z}\))(n ≥ a =⇒ P (n)), where P (n) is some predicate and a ∈ \(\mathbb{Z}\).
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/05%3A_New_Page
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/02%3A_New_Page/2.05%3A_New_PageNote that it would not be appropriate to utilize the “without loss of generality” approach to combine the two cases in the proof of Theorem 2.88 since the proof of the second case is not as simple as ...Note that it would not be appropriate to utilize the “without loss of generality” approach to combine the two cases in the proof of Theorem 2.88 since the proof of the second case is not as simple as swapping the roles of symbols in the proof of the first case.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/04%3A_New_Page
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/01%3A_New_Page/1.04%3A_New_PageSome Problems are computational in nature and aimed at improving your understanding of a particular concept while others ask you to provide a counterexample for a statement if it is false or to provid...Some Problems are computational in nature and aimed at improving your understanding of a particular concept while others ask you to provide a counterexample for a statement if it is false or to provide a proof if the statement is true. The overarching goal of this book is to help you develop a deep and meaningful understanding of the processes of producing mathematics by putting you in direct contact with mathematical phenomena.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/02%3A_New_Page/2.02%3A_New_Page(e) The proposition “A if and only if B” (alternatively, “A is necessary and sufficient for B”) is true if both A and B have the same truth value; expressed symbolically as \(A ⇐⇒ B\) and called a bic...(e) The proposition “A if and only if B” (alternatively, “A is necessary and sufficient for B”) is true if both A and B have the same truth value; expressed symbolically as \(A ⇐⇒ B\) and called a biconditional proposition. It is worth pointing out that definitions in mathematics are typically written in the form “B if A” (or “B provided that A” or “B whenever A”), where B contains the term or phrase we are defining and A provides the meaning of the concept we are defining.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/04%3A_New_Page/4.04%3A_New_PageIn other words, one can prove the Well-Ordering Principle from the Axiom of Induction, as we have done, but one can also prove the Axiom of Induction if the Well-Ordering Principle is assumed. If \(A\...In other words, one can prove the Well-Ordering Principle from the Axiom of Induction, as we have done, but one can also prove the Axiom of Induction if the Well-Ordering Principle is assumed. If \(A\) is a nonempty subset of the integers and there exists \(\ell\in \mathbb{Z}\) such that \(\ell\leq a\) for all \(a\in A\), then \(A\) contains a least element.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/01%3A_New_Page
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/06%3A_New_Page/6.01%3A_New_PageThe Fundamental Theorem of Arithmetic (sometimes called the Unique Factorization Theorem) states that every natural number greater than 1 is either prime or is the product of prime numbers, where this...The Fundamental Theorem of Arithmetic (sometimes called the Unique Factorization Theorem) states that every natural number greater than 1 is either prime or is the product of prime numbers, where this product is unique up to the order of the factors. Now, assume that \(d>1\) and define \[S:= \{n-dk\mid k\in\mathbb{Z}\text{ and } n-dk\geq 0\}.\] If we can show that \(S\neq\emptyset\), then we can apply the Well-Ordering Principle (Theorem 4.38) to conclude that \(S\) has a least element of S.
