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| Question | Answer |
| Definition 1: What is a RELATION? | A relation, R, on a non-empty set S, is a non-empty SUBSET of the CARTESIAN PRODUCT; S X S |
| Definition 2: What is an EQUIVALENCE RELATION? | A relation, R, on a non-empty set S is said to be an equivalence relation if the following are satisfied: 1) Reflexive: aRa, for all a in S 2) Symmetric: aRb => bRa, for all a,b in S 3) Transitive: aRb, bRc => aRc, for all a,b,c in S |
| Definition 3: What is a PARTITION? | A partition of a set S is a collection, P, of non-empty subsets of S s.t. every element of S is in EXACTLY one member of P ie. [a]=R(a)={x in A|xRa} NOTE: the Partition exhausts set |
| Definition 4: What is an EQUIVALENCE CLASS? | Each of the members of a partition is called an equivalence class |
| Definition 5: What is a BINARY OPERATION? | A binary operation on a set S, is an operation * s.t. x*y is UNIQUELY DEFINED for all x, y in S. It must satisfy: 1) Closure: x*y is also in S 2) Associativity: (x*y)*z=x*(y*z) for all x,y,z in S 3) Commutativity: x*y=y*x for x,y in S 4) Identity: there exists an element, e, in S w.r.t.* iff, e*x=x*e=x |
| Definition 6: What is an INVERSE? | Let S be a set, closed under binary operation * with identity e. An inverse of x, is any element, y in S s.t. x*y=y*x=e. Generally, the inverse of x is denoted x^-1 |
| Definition 7: What is the definition of powers of operations? | Let S be a set closed under an ASSOCIATIVE binary operation *, then x^n=x*x*x*...*x (n terms) where n is a positive integer Note: It then follows that the "usual" index laws apply, x^0 = e and x^-n=(x^-1)^n |
| Definition 8: What is a GROUP? | A group is a pair (S, *) where S is a non-empty set and * is a binary operation defined on S, s.t.: 1) S is closed under * 2) * is associative on S 3) S has an identity element w.r.t x denoted e where e is also an element of S 4) Every element of S has an inverse also in S |
| Definition 9: What does ABELIAN mean? | A group (G, *) is called Abelian (OR COMMUTATIVE) iff for all x and y in G: x*y=y*x (so a group plus Commutativity) |
| Definition 10: What is an ISOMETRY? | An Isometry of the plane is a BIJECTIVE (injective and surjective) maping which preserves distances |
| Definition 11: What is a MATRIX GROUP? | Let GL(n, Reals) be the set of all non-singular (ie. invertible) matrices in Reals_nxn. Then: (GL(n, Reals),X) is a group called the General Linear Group of Degree n The X is multiplication in this case |
| Definition 12: What is a SYMMETRY GROUP? | Let P be the set of points representing a figure in space and G be the set of all isometries which map P onto ITSELF (called the symmetries of P). Then, (G, *) is a group called the Symmetry Group of P. |
| Definition 13: What does the word ORDER refer to? | The order of a SET, S, denoted |S| is the number of elements in S. If S is an infinite set we shall write |S| = infinity |
| Definition 14: What does the PERIOD of an element mean? | Let (G, *) be a group with identity e in G and consider some a in G. If, for all n (positive integer) a^n does NOT equal e, then a^e has infinite period. Otherwise, it has period k s.t. a^k=e where k is the smallest k as positive integer. NOTE: a^3= a*a*a where k=3 |
| Definition 15: What is a finite subgroup? | Let H be a non-empty, finite SUBSET of the elements of a group (that is not necessarily finite) (G,*). Then, (H,*) is a subgroup of (G,*) IFF H is CLOSED under * |
| Definition 16: Define a CYCLIC SUBGROUP | Let (G, *) be a group and A be in G. Then: {A^n|n is an integer} forms a subgroup of G called the cyclic subgroup generated by A denoted <A>. |
| Definition 17: What is a CYCLIC GENERATOR? | A group, G, is called cyclic IFF there exists an A in G called a generator of G s.t. G=<A> which generates every element of the group |
| Definition 18: What is the definition of a LEFT COSET? | Let (H,*) be a subgroup of (G,*) note: same operation For a fixed element g in G, the set: gH={gh|h is an element of H} is the left coset of H in G. PRE-MULTIPLICATIONS If H is finite, then gH={g*h1, g*h2,...g*hn| n is a positive integer} |
| Definition 19: What is an ISOMETRY? | Let (G1, *) and (G2, o) be groups. Then G1 and G2 are isomorphic IFF there exists a mapping f:G1-> G2 (called an isomorphism) s.t. 1) f is bijective (injective and surjective) 2) f(x*y)=f(x) o f(y), for all x,y in G1 (Isomorphic has notation like congruence) |
| Definition 20: What is a HOMOMORPHISM? | Let G1 and G2 be groups. A mapping h: G1->G2 is a group homomorphism IFF, for all x, y in G1: h(xy)=h(x)h(y) NOTE: This is a surjective ISOMORPHISM |
| Definition 21: Define an Image and, similarly, the Preimage. | Let f be a mapping of a set X into a set Y and let A be a subset of X and B be a subset of Y. Then, 1) the IMAGE of A under f is a subset of Y and is denoted and defined by: f(A)={f(a)|a in A} 2) the PREIMAGE of B under f is a subset of X and is denoted and defined by: f^-1(B)={x in X|f(x) is in B} |
| Definition 22: What is a Kernal? | Let # be a group homomorphism from a group into a group G' and let e' be the identity of G'. The Kernal of # is denoted and defined by: Ker(#)={x in G|#(x)=e'} |
| Definition 23: What does the term PERMUTATION refer to? | A permutation of a finite set A is a function f:x->x that is bijective |
| Definition 24: What is a SYMMETRIC group? | Let X be the finite set {1,2,...n}. The group of all permutitions of X is the symmetric group on n-items and is denoted Sn NOTE: Sn has n! elements |
| Definition 25: What is CYCLE NOTATION? | In general, f=(a_1 a_2 a_3 ... a_m-1 a_m) is used to denote the permutation: f(a_1)=a_2, f(a_2)=a_3, .... f(a_m-1)=a_m, f(a_m)=a_1 |
| Definition 26: What is a DISJOINT CYCLE? | Two cycles are said to be DISJOINT if they do not have any common elements |
| Definition 27: What is an ORBIT? | The disjoint cycles of a permutation, including SINGLETONS (one element) are called the orbits of a permutation. |
| Definition 28: What does the term TRANSPOSITION refer to? | A cycle (i,j) of length TWO INTERCHANGES i and j is called a transposition. NOTE: any cycle can be expressed as a product of transpositions |
| Definition 29: What does PARITY mean? | A permutation is odd or even according to whether it can be expressed as a product of an odd or even number of transpositions respectively |
| Definition 30: What is the ALTERNATING GROUP? | The SET of all EVEN permutations of degree n for a subgroup of Sn called the Alternating Group of Degree n and is denoted An. |
| Definition 31: What is the CARTESIAN PRODUCT OF SETS? | Let S1, S2, ... Sn be sets. Then, the Cartesian product for these sets is the set of all n-tuples (a1, a2, a3, ... an) where ai is an element of Si for all i =1,2,...n We denote the Cartesian product as: S1 X S2 X S3 X... X Sn where X means "cross" |
| Definition 32: Define a PRIME NUMBER | A p (positive integer) is prime iff, p>1 and the only positive divisors of p are 1 and p itself |
| Definition 33: What is a RESIDUE CLASS? | Let A and n be integers (n positive). The residue class [A]n of A modulo n is the EQUIVALENCE CLASS containing A under congruence modulo n; that is: [A]n={m|m is an integer, m congruent to A(modn)} = {A+kn|k is an integer} NOTE: Zn (integer addition modulo n)= {[0]n,[1]n,..., [n-1]n} |
| Definition 34: Describe the REDUCED RESIDUE SYSTEM | The Reduced Residue System modulo n is the set denoted and defined by: Zn^x (integer)={[a]n in Zn (integer)Z there exists [b]n in Zn (integer) s.t. [a]n[b]n=[1]n} where [1]n is the multiplicative identity Which implies that gcd(a,n)=1 (ie. relatively prime) |
| Definition 35: What is the EULER TOTIENT FUNCTION? | Let m,n be natural numbers. Then the Euler Totient Function denoted y is: y: N(natural)-> N(natural) defined by: y(n)=|{m is natural s.t. m<_ n and gcd(m,n)=1}| for all n (natural numbers) THIS IS ORDER This is the order of Zn^x (integers) |
| Definition 36: What is a RING? | A ring is a triple consisting of a non-empty set S equipped with two binary operations denoted + and X s.t: 1) + holds under closure, associativity, identity, inverse and commutativity 2) X holds under closure, associativity and the distributive NOTE: with a one is the special property of a multiplicative identity |
| Definition 37: What is a SUBRING? | Let R be a ring and let S be a subset of R. Then, S is a subring of R if the following are satisfied: 1) S has an identity element 2) a and b in S means a minus b is in S 3) a and b in S means that ab is in S e.g. {0_R} the additive identity is a subring and R itself is the trivial subring |
| Definition 38: What is the RING KERNAL? | Let R1 and R2 be rings and let f:R1->R2 be a ring homomorphism. The kernal is the SET: ker(f) = {x in R1| f(x)=0_R2} |
| Definition 39: What is meant by the phrase 'A DIVISOR OF ZERO'? | Let R be a non-zero ring (Excluding the additive identity {0_R}) with a multiplicative inverse and let x be in R and nonzero. We say that x is a divisor of zero if there exists a y in R without the additive identity s.t. xy=0_R |
| Definition 40: What is an INTEGRAL DOMAIN? | An integral domain is a COMMUTATIVE, nonzero ring with a multiplicative identity in which there are no divisors of 0 |
| Definition 41: What is a UNIT? | Let R be a ring. If A in R has a multiplicative inverse in R, then A is said to be a unit in R. |
| Definition 42: What is a DIVISION RING? | A division ring is a ring (not necessarily commutative) in which every non-zero element is a unit. Hence, for all A in R without 0_R, there exists A^-1 s.t. AxA^-1=A^-1xA=1+R |
| Defintion 43: What is a FIELD? | A field is a COMMUTATIVE division ring |
| Definition 44: What is the CHARACTERISTIC of a ring? | Let R be an Integral domain. The characteristic of R, denoted char(R), is the period of 1_R in the group (G, +). If 1_R has infinite period, we write: char(R)=infinity |
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