Lecture Notes 2
... In some cases it’s easier to prove opposite to inverse theorem than the direct theorem. If a theorem has a structure of an implication x → y, then x is called the sufficient condition of the theorem, while y is its necessary condition. If a theorem has a form “x is necessary and sufficient for y” it mea ...
... In some cases it’s easier to prove opposite to inverse theorem than the direct theorem. If a theorem has a structure of an implication x → y, then x is called the sufficient condition of the theorem, while y is its necessary condition. If a theorem has a form “x is necessary and sufficient for y” it mea ...
A + B + C
... In POS standard form, every variable in the domain must appear in each sum term of the expression. You can expand a nonstandard POS expression to standard form by adding the product of the missing variable and its complement and applying rule 12, which states that (A + B)(A + C) = A + BC. Convert X ...
... In POS standard form, every variable in the domain must appear in each sum term of the expression. You can expand a nonstandard POS expression to standard form by adding the product of the missing variable and its complement and applying rule 12, which states that (A + B)(A + C) = A + BC. Convert X ...
Intermediate Value Theorem (IVT)
... Warm Up: Refresh your memory on what the complex zero theorem says then use it to answer the example question. ...
... Warm Up: Refresh your memory on what the complex zero theorem says then use it to answer the example question. ...
Lesson 1
... – Select a number. Multiply the number by 6. Add 8 to the product. Divide this sum by 2. Subtract 4 from the quotient. – Repeat this procedure for 4 different numbers and write a conjecture – Represent the original number by n and use deductive reasoning to prove the conjecture ...
... – Select a number. Multiply the number by 6. Add 8 to the product. Divide this sum by 2. Subtract 4 from the quotient. – Repeat this procedure for 4 different numbers and write a conjecture – Represent the original number by n and use deductive reasoning to prove the conjecture ...
All of Math in Three Pages
... Here is an overview of “high school mathematics.” Most of the non-Calculus topics that you have studied and a lot that you haven’t will appear somewhere in the following outline. In a some cases a formula or example is given, but many of these important results are just named, which leaves the inter ...
... Here is an overview of “high school mathematics.” Most of the non-Calculus topics that you have studied and a lot that you haven’t will appear somewhere in the following outline. In a some cases a formula or example is given, but many of these important results are just named, which leaves the inter ...
[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments (§2.3
... A formal proof of a conclusion C, given premises p1, p2,…,pn consists of a sequence of steps, each of which applies some inference rule to premises or previously-proven statements (antecedents) to yield a new true statement (the consequent). A proof demonstrates that if the premises are true, then t ...
... A formal proof of a conclusion C, given premises p1, p2,…,pn consists of a sequence of steps, each of which applies some inference rule to premises or previously-proven statements (antecedents) to yield a new true statement (the consequent). A proof demonstrates that if the premises are true, then t ...
MATH 251
... - Recognize and use various types of reasoning and methods of proof -How to use abstract algebraic structures in solving problems expressed by symbols 3. Professional and practical skills: - Use combinatorial techniques when needed in solving problems. - Use of logic to determine the validity of an ...
... - Recognize and use various types of reasoning and methods of proof -How to use abstract algebraic structures in solving problems expressed by symbols 3. Professional and practical skills: - Use combinatorial techniques when needed in solving problems. - Use of logic to determine the validity of an ...
Mathematics in Context Sample Review Questions
... From this triangle construct two squares with sides of length a + b, also shown above. The two squares have the same lengths for their sides, so their areas must be equal. a. Calculate the area of the left-hand square by adding up the areas of the triangles and squares that compose it. (4) ...
... From this triangle construct two squares with sides of length a + b, also shown above. The two squares have the same lengths for their sides, so their areas must be equal. a. Calculate the area of the left-hand square by adding up the areas of the triangles and squares that compose it. (4) ...
Proof
... • “If n is odd and m ≡ 3 (mod 4), then (n2 + m) is divisible by 4.” (More complicated than midterm.) ...
... • “If n is odd and m ≡ 3 (mod 4), then (n2 + m) is divisible by 4.” (More complicated than midterm.) ...
The Origin of Proof Theory and its Evolution
... mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithmetic is a first-order theory commonly formalized independently in first-order logic. It constitutes a fundamental formalism for arithmetic, and ...
... mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithmetic is a first-order theory commonly formalized independently in first-order logic. It constitutes a fundamental formalism for arithmetic, and ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.