Algebra Volume 2, 2nd Edition by P. M. Cohn

By P. M. Cohn

The most emphasis of this revised algebra textbook is on fields, earrings and modules. The textual content contains new chapters at the consultant conception of finite teams, coding idea and algebraic language idea. units, lattices, different types and graphs are brought first and foremost of the textual content. The textual content, which has been rewritten with the purpose of constructing the topic more uncomplicated to know, comprises simplified proofs and lots of new illustrations and workouts.

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In situations like this it is customary to use slightly different notations for the union and intersection of the family. For instance, instead of writing A∈A A we write ∞ An or An . n=0 n∈N We can use a variation on this latter notation to denote the union of just some of the sets, like 5 An n=3 for the union of the family {A3 , A4 , A5 }. 5 For n ∈ N, put An = [n, n + 2). We have Example ∞ ∞ 5 An = [0, ∞), n=0 3 An = ∅, An = [3, 7), n=3 An = [3, 4). n=0 n=2 Discussion: These equations are arrived at by picturing the intervals An on the number line as n ranges as indicated.

Prove that X ⊆ (−3, 1). Proof Let x ∈ X. Then x = 1 − r for some r ∈ [1, 3]. First, r ∈ R, so x = 1 − r ∈ R, which is one requirement for x to be in (−3, 1). Now, since r ≥ 1, we have −r ≤ −1 so that x = 1 − r ≤ 0 < 1. And since r ≤ 3, we have −r ≥ −3 so that x = 1 − r ≥ −2 > −3. Therefore, x ∈ (−3, 1) and we conclude that X ⊆ (−3, 1). 5 Example Define f : R → R by f (x) = 2 − 3x and put A = {x ∈ R | f (x) ∈ [−10, 5)}. Prove that A ⊆ (−2, 6). Proof Let x ∈ A. Then x ∈ R and f (x) ∈ [−10, 5). This latter condition says −10 ≤ f (x) < 5, that is, −10 ≤ 2 − 3x < 5.

Since this is a for-every statement, we write “Let A0 ∈ A” (using the notation A0 so that we can continue to use A for a dummy variable). Now the task is to show that x ∈ X ∪ A0 , which we do by assuming x ∈ / X and showing x must be in A0 . In the proof of the containment (⊇), after letting x be an arbitrary element of the right-hand side and writing what this means we turn to the task of showing that x must be an element of the left-hand side, that is, x ∈ X or x ∈ A∈A A. 1), that is, x ∈ A for every A ∈ A.

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