Abstract Algebra 1

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  1. Definition of the Cartesian product of two sets.
    Let A and B be sets. The set A x B = {(a,b) | a Image Upload 1 A and b Image Upload 2 B} is the cartesian product of A and B.
  2. Definition a function and related concepts.
    A function f mapping X onto Y is a relation between X and Y with the property that each x Image Upload 3 X appears as the first member of exactly one ordered pair (x,y) in f. Such a function is also called a map of mapping of X into Y. We write f:X->Y and express (x,y) Image Upload 4 f by f(x)=y. The domain of f is the set X and the set Y is the codomain of f. The range of f is f[X] = {f(x) | xImage Upload 5 X}.
  3. Definition of injective (one-to-one), surjective (onto), and bijective function.
    A function f:X->Y is one to one if f(x1)=f(x2) only when x1=x2. The function f is onto Y if the range of f is Y. The function f is bijective if it is one to one and onto.
  4. Definition of two sets with the same cardinality.
    Two sets X and Y have the same cardinality if there exists a one-to-one function mapping X onto Y, that is, if there exists a one-to-one correspondence between X and Y.
  5. Prove that the sets Z and Q have cardinality Image Upload 6.
    For each n Image Upload 7 N, let An be the set given by An = {Image Upload 8p/q | where p,q Image Upload 9 N are in lowest terms with p + q = n}. Each An is finite and every rational number appears in exactly one of these sets. A countable union of finite sets is countable.
  6. Definition of partition.
    A partition of a set S is a collection of nonempty subsets of S such that every element of S is in exactly one of the subsets. The subsets are the cells of the partition.
  7. Definition of an equivalence relation.
    • An equivalence relation R on a set S is one that satisfies these three properties for all x, y, z Image Upload 10 S.
    • 1. (Reflexive) xRx.
    • 2. (Symmetric) If xRy, then yRx.
    • 3. (Transitive) If xRy and yRz, then xRz.
  8. Define Congruuence Modulo n.
    Let n Image Upload 11 Z+. The equivalence relation on Z+ corresponding to the partition of Z+ into residue classes modulo n is congruence modulo n. It is denoted a =n b or a = b (mod 4), read "a is congruent to b modulo n."
  9. (Equivalence Relations and Partitions) Let S be a nonempty set and let ~ be an equivalence relation on S. Then ~ yields a partition of S, where Image Upload 12 = {x Image Upload 13 S | x ~ a}. Also, each partition of S gives rise to an equivalence relation ~ on S where a ~ b if and only if a and b are in the same cell of the partition.
    We must show that the different cells Image Upload 14 = {x Image Upload 15 S | x ~ a} for a Image Upload 16 S do give a partition of S, so that every element of S is in some cell and so that if a Image Upload 17 Image Upload 18, then Image Upload 19 = Image Upload 20. Let a Image Upload 21 S. Then a Image Upload 22 Image Upload 23 by the reflexive condition, so a is in at least one cell. Suppose now that a were in a cell Image Upload 24 also. We need to show that Image Upload 25 = Image Upload 26 as sets; this will show that a cannot be in more than one cell. There is a standard way to show that two sets are the same: Show that each set is a subset of the other. We show that Image Upload 27 Image Upload 28 Image Upload 29. Let x Image Upload 30 Image Upload 31. Then x ~ a. But a Image Upload 32 Image Upload 33, so a ~ b. Then, by the transitive condition, x ~ b, so x Image Upload 34 Image Upload 35. Thus Image Upload 36 Image Upload 37 Image Upload 38. Now we show that Image Upload 39 Image Upload 40 Image Upload 41. Let y Image Upload 42 B. Then y ~ b. But a Image Upload 43 Image Upload 44, so a ~ b and by symmetry b ~ a. Then by transitivity y ~ a, so y Image Upload 45 Image Upload 46. Hence Image Upload 47 Image Upload 48 Image Upload 49 also, so Image Upload 50 = Image Upload 51 and our proof is complete.
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Abstract Algebra 1

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