Introduction to Abstract Algebra, Second Edition presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It avoids the usual groups first/rings first dilemma by introducing semigroups and monoids, the multiplicative structures of rings, along with groups.This new edition of a widely adopted textbook covers
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Yes, you can access Introduction to Abstract Algebra by Jonathan D. H. Smith in PDF and/or ePUB format, as well as other popular books in Mathematics & Algebra. We have over one million books available in our catalogue for you to explore.
Algebra begins as the art of working with numbers. The integers are the whole numbers, positive, negative, and zero. Put together, they form the set
(the letter ℤ coming from the German word
, meaning “numbers”). The natural numbers are the nonnegative integers, including zero. They are “natural” because they are the possible numbers of elements in a finite set. For example, 4 is the number of elements of the set
of suits in a deck of cards, while 13 is the number of elements of the set
of cards in the suit ♡ of hearts. Note that 0 is the number of elements in the empty set ∅ or { }. The natural numbers form the set
Another set of numbers familiar from calculus is the set ℝ of real numbers, like −17, √2 = 1.41421..., e = 2.71828 ..., π = 3.14159 ..., and so on. It is hard to display the set of real numbers as a list of elements between braces, like the sets (1.1)-(1.4) above. Instead, the set ℝ is pictured as the real line
(like an axis in the graph of a function). Pictures like this are useful as geometric visualizations of real numbers. At times similar pictures can even be useful for natural numbers or integers, since these numbers also happen to be real numbers.
1.1 Ordering numbers
In calculus, order relations between real numbers are crucial, for instance when we want to find the maximum value of a function over a certain range. Recall that x < y (read “x less than y”) means y − x is positive, while x ≤ y (read “x less than or equal to y”) means that y − x is nonnegative. We can also write y > x (“y greater than x”) instead of x < y, or y ≥ x (“y greater than or equal to x”) instead of x ≤ y. In the real line picture, with the positive numbers going off to the right, the relation x < y becomes an arrow x → y. It is often helpful to signify the relation x ≤ y with an arrow from x to y, without requiring the arrow to go horizontally from left to right.
Since algebra also needs to work with order relations between numbers, it is important to know the rules for manipulating them. The first rule is called reflexivity:
