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# 数学代写|抽象代数代写Abstract Algebra代考|MATH413 Theorems and proofs in Abstract Algebra

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## 数学代写|抽象代数代写Abstract Algebra代考|Theorems and proofs in Abstract Algebra

irst things first: if you flipped straight to this chapter because you are struggling with theorems and proofs in Abstract Algebra, please start with Chapter 2. People often work on theorems and proofs without fully understanding the underlying axioms and definitions, which makes everything more difficult. If you have read Chapter 2 , you will know that I think of mathematical theories as in the diagram below, where the ‘bottom’ layer contains axioms and definitions. Theorems are proved from these axioms and definitions; they are the ‘results’ of the deductive science that is mathematics.

But what does it mean to prove theorems or to say that mathematics is deductive? A deductive argument is one in which the conclusion is a necessary consequence of the premises (also called assumptions or hypotheses). You might not have thought of mathematics this way, but you are nevertheless accustomed to deductive reasoning. The algebraic step from ‘ $(x-2)(x-5)=0$ ‘ to ‘ $x=2$ or $x=5$ ‘ is a deduction: if the premise that $(x-2)(x-5)=0$ is true, it necessarily follows that $x=2$ or $x=5$. A proof chains such deductions together: each step introduces relevant objects or can be justified using axioms, definitions and earlier results. Proofs can be long and complicated, but they can also be short and simple, like this.

## 数学代写|抽象代数代写Abstract Algebra代考|Logic in familiar algebra

This section reviews some familiar algebra with a focus on its underlying logic. You might read it and think, yeah, I know all that. But many readers will recognize that while they ‘know’ it in the sense that they could act accordingly, they have not systematically reflected upon this knowledge. This review will build on the discussion in Section 1.2, considering logic in algebra in relation to numbers and matrices.

First, consider equation solving. What would you say it means to solve an equation? Can you get beyond ‘finding $x$ ‘? Maybe think about how you would explain equation solving to a young student who is intelligent but has not yet studied equations. What would you say? I will ask again later. Perhaps the simplest equations take forms as in Section $1.3: x+a=b$ or $a x=b$. If these seem trivial, that is due to your extensive knowledge. For young children, the world of numbers is smaller than it is for you, and the equations $x+5=2$ and $5 x=2$ have no solutions. For you, they do have solutions because you know about negative and rational numbers. In Abstract Algebra, we do not revert to the earlier position, but nor do we assume that all numbers are always fair game-we are careful about sets. In the integers, $5 x=2$ has no solution. In the real numbers, $x^2=-5$ has no solution.

As discussed in Section 1.3, to guarantee that all equations of the form $x+a=b$ can be solved requires that the manipulations below are valid. So it requires a set that is closed under addition and that has associative addition, an additive identity and additive inverses. In short, it requires an additive group.

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