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Abstract algebra (Binary operations (Sets (General sets (Z represents the…
Abstract algebra
Binary operations
Sets
Collection of individual objects called elements
Finite or infinite
General sets
N represents the set of natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9}
Z represents the set of integers, positive or negative, including 0
Z- is the set of all negative integers
Z+ is the set of positive integers
Q represents all rational numbers, numbers that can be written in the form a/b, where a and b are real
R represents all real numbers, including those which are irrational and transcendental
C represents all numbers that have both an imaginary part and a real part
A function is a binary operation if it can be applied to any two elements of a set so that the result is also a member of the set
A binary operation,
*, is commutative if a
b = b*a for all a and b
A binary operation, ☆, is associative if (a☆b)☆c = a☆(b☆c)
The identitiy element, e, of a set under an operation ☆ is such that a☆e = e☆a = a for all values in the set
The identity element is unique
The inverse, a^-1, of an element a under an operation ☆ is such that a☆a^-1 = a^-1☆a = e
An element, a, is a self-inverse if a^-1 = a
Cayley tables
Named after the nineteenth-century mathematician Arthur Cayley
A table showing the result of a binary operation on all possible pairs of elements
Strategy
1 Look along the columns to find one the same as the initial column of elements; the element at the top of the column is the identity element, e. Alos check along the relevant row to make sure it equals the initial row of elements
2 Find all the instances of e in the table
3 Work down the rows. For each element, a, at the start of the row the element at the top of the column containing e is the inverse of a
Modular arithmetic
If a ≡ b (mod n) and c ≡ d (mod n) then:
a+c = b+d (mod n)
a-c = b-d (mod n)
ac = bd (mod n) as long as a, b, c, d, n are integers
Strategy
1 Draw the table for the set {0, 1, 2,..., (n-1)}
2 Calculate a*b for each pair of integers and write the answers (mod n) in the tables
3 Use the fact that a
e = e
a for the identity element e
The notation +n represents addition modulo n, where n is a positive integer
The notation Xn represents multiplication modulo n, where n is a positive integer