...George Moore
Seven has always been considered a special number. To the Egyptians, for example, the earth was represented by a four-sided house in which three gods dwelled, which added up to seven. This became their lucky number.
Seven really came into its own with the Christian interpretation of Creation:
The world was made in seven days.
There are seven days to the week.
There are seven graces.
There are seven stanzas in the Lord�s Prayer.
There are seven ages of man.
Christ uttered seven last words.
Most of the above beliefs are ruled by the different phases of the moon, which change every seven days. The Romans believed that the mind and the body changed completely and were renewed after seven years. They also started the seven year�s bad luck concept.
Seventh-Heaven is an Islamic concept, and it represents the best of all possible places. It is the heaven of heavens, the residence of God and His angels. There is also a very early Islamic belief that the there are seven heavens, one lying right above the other, graduating in degrees; depending upon how good a person was on earth, he or she could say, �I�m in Seventh Heaven�.
Seven is especially lucky to gambles.
The seventh son born to the seventh son is thought to be doubly blessed. He is believed to be a clairvoyant, with natural healing powers. Throughout the Middle Ages, the seventh son usually practiced magic and administered the laying on of hands to the sick.
A mother and father have six sons and each son has one sister. How many people are in that family?
1.1 Sets of Numbers and Interval Notation
1.2 Operations on Real Numbers
1.3 Simplifying Expressions
1.4 Linear Equations in One Variable
1.5 Applications of Linear Equations in One Variable
1.6 Literal Equations and Applications to Geometry
1.7 Linear Inequalities in One Variable
1.8 Properties of Integer Exponents and Scientific Notation
2.1 The Rectangular Coordinate System and Midpoint Formula
2.2 Linear Equations in Two Variables
2.3 Slope of a Line
2.4 Equations of a Line
2.5 Applications of Linear Equations and Graphing
3.1 Solving Systems of Linear Equations by Graphing
3.2 Solving Systems of Equations by Using the Substitution Method
3.3 Solving Systems of Equations by Using the Addition Method
3.4 Applications of Systems of Linear Equations in Two Variables
3.5 Systems of Linear Equations in Three Variables and Applications
3.6 Solving Systems of Linear Equations by Using Matrices
3.7 Determinants and Cramer�s Rule
4.1 Introduction to Relations
4.2 Introduction to Functions
4.3 Graphs of Functions
4.4 Variation
5.1 Addition and Subtraction of Polynomials and Polynomial Functions
5.2 Multiplication of Polynomials
5.3 Division of Polynomials Problem Recognition Exercises�Operations on Polynomials
5.4 Greatest Common Factor and Factoring by Grouping
5.5 Factoring Trinomials
5.6 Factoring Binomials
5.7 Additional Factoring Summary
5.8 Solving Equations by Using the Zero Product Rule
6.1 Rational Expressions and Rational Functions
6.2 Multiplication and Division of Rational Expressions
6.3 Addition and Subtraction of Rational Expressions
6.4 Complex Fractions Problem Recognition Exercises�Operations on Rational Expressions
6.5 Rational Equations
6.6 Applications of Rational Equations and Proportions
7.2 Rational Exponents
7.3 Simplifying Radical Expressions
7.4 Addition and Subtraction of Radicals
7.5 Multiplication of Radicals
7.6 Rationalization
7.7 Radical Equations
7.8 Complex Numbers
8.2 Quadratic Formula
8.3 Equations in Quadratic Form
8.4 Graphs of Quadratic Functions
8.5 Vertex of a Parabola and Applications
9.2 Polynomial and Rational Inequalities
9.3 Absolute Value Equations
9.4 Absolute Value Inequalities Problem Recognition Exercises�Equations and Inequalities
9.5 Linear Inequalities in Two Variables
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2.3.1. Definition. Let S be a set. A function :S->S is called a permutation of S if is one-to-one and onto.
The set of all permutations of S will be denoted by Sym(S).
The set of all permutations of the set { 1, 2, ..., n } will be denoted by Sn.
Proposition 2.1.6 shows that the composition of two permutations in Sym(S) is again a permutation. It is obvious that the identity function on S is one-to-one and onto.
Proposition 2.1.8 shows that any permutation in Sym(S) has an inverse function that is also one-to-one and onto. We can summarize these important properties as follows:
(i) If , Sym(S), then Sym(S);
(ii) 1S Sym(S);
(iii) if Sym(S), then -1 Sym(S).
2.3.2. Definition. Let S be a set, and let Sym(S). Then is called a cycle of length k if there exist elements a1, a2, ..., ak S such that
(a1) = a2, (a2) = a3, . . . , (ak-1) = ak, (ak) = a1, and
(x)=x for all other elements x S with x ai for i = 1, 2, ..., k.
In this case we write = (a1,a2,...,ak).
We can also write = (a2,a3,...,ak,a1) or = (a3,...,ak,a1,a2), etc. The notation for a cycle of length k can thus be written in k different ways, depending on the starting point. The notation (1) is used for the identity permutation.
2.3.3. Definition. Let = (a1,a2,...,ak) and = (b1,b2,...,bm) be cycles in Sym(S), for a set S. Then and are said to be disjoint if ai bj for all i,j.
2.3.4. Proposition. Let S be any set. If and are disjoint cycles in Sym(S), then = .
2.3.5. Theorem. Every permutation in Sn can be written as a product of disjoint cycles. The cycles that appear in the product are unique.
2.3.6. Definition. Let Sn. The least positive integer m such that m = (1) is called the order of .
2.3.7. Proposition. Let Sn have order m. Then for all integers j,k we have
j = k if and only if j k (mod m).
2.3.8. Proposition. Let Sn be written as a product of disjoint cycles. Then the order of is the least common multiple of the lengths of its cycles.
2.3.9. Definition. A cycle (a1,a2) of length two is called a transposition.
2.3.10 Proposition. Any permutation in Sn, where n 2, can be written as a product of transpositions.
2.3.11. Theorem. If a permutation is written as a product of transpositions in two ways, then the number of transpositions is either even in both cases or odd in both cases.
2.3.12. Definition. A permutation is called even if it can be written as a product of an even number of transpositions, and odd if it can be written as a product of an odd number of transpositions.
3.1.3. Definition. A group (G,�) is a nonempty set G together with a binary operation � on G such that the following conditions hold:
(i) Closure: For all a,b G the element a � b is a uniquely defined element of G.
(ii) Associativity: For all a,b,c G, we have
a � (b � c) = (a � b) � c.
(iii) Identity: There exists an identity element e G such that
e � a = a and a � e = a
for all a G.
(iv) Inverses: For each a G there exists an inverse element a-1 G such that
a � a-1 = e and a-1 � a = e.
We will usually simply write ab for the product a � b.
3.1.6. Proposition. (Cancellation Property for Groups) Let G be a group, and let a,b,c G.
(a) If ab=ac, then b=c.
(b) If ac=bc, then a=b.
3.1.8. Definition. A group G is said to be abelian if ab=ba for all a,b G.
3.1.9. Definition. A group G is said to be a finite group if the set G has a finite number of elements. In this case, the number of elements is called the order of G, denoted by |G|.
3.2.7. Definition. Let a be an element of the group G. If there exists a positive integer n such that an = e, then a is said to have finite order, and the smallest such positive integer is called the order of a, denoted by o(a).
If there does not exist a positive integer n such that an = e, then a is said to have infinite order.
3.2.1. Definition. Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G.
3.2.2. Proposition. Let G be a group with identity element e, and let H be a subset of G. Then H is a subgroup of G if and only if the following conditions hold:
(i) ab H for all a,b H;
(ii) e H;
(iii) a-1 H for all a H.
3.2.10. Theorem. (Lagrange) If H is a subgroup of the finite group G, then the order of H is a divisor of the order of G.
3.2.11. Corollary. Let G be a finite group of order n.
(a) For any a G, o(a) is a divisor of n.
(b) For any a G, an = e.
Example 3.2.12. (Euler's theorem) Let G be the multiplicative group of congruence classes modulo n. The order of G is given by (n), and so by Corollary 3.2.11, raising any congruence class to the power (n) must give the identity element.
3.2.12. Corollary. Any group of prime order is cyclic.
3.4.1. Definition. Let G1 and G2 be groups, and let : G1 -> G2 be a function. Then is said to be a group isomorphism if
(i) is one-to-one and onto and
(ii) (ab) = (a) (b) for all a,b G1.
In this case, G1 is said to be isomorphic to G2, and this is denoted by G1 G2.
3.4.3. Proposition. Let : G1 -> G2 be an isomorphism of groups.
(a) If a has order n in G1, then (a) has order n in G2.
(b) If G1 is abelian, then so is G2.
(c) If G1 is cyclic, then so is G2.
Cyclic groups
3.2.5 Definition. Let G be a group, and let a be any element of G. The set
= { x G | x = an for some n Z }
is called the cyclic subgroup generated by a.
The group G is called a cyclic group if there exists an element a G such that G=. In this case a is called a generator of G.
3.2.6 Proposition. Let G be a group, and let a G.
(a) The set is a subgroup of G.
(b) If K is any subgroup of G such that a K, then K.
3.2.8. Proposition. Let a be an element of the group G.
(a) If a has infinite order, and ak = am for integers k,m, them k=m.
(b) If a has finite order and k is any integer, then ak = e if and only if o(a) | k.
(c) If a has finite order o(a)=n, then for all integers k, m, we have
ak = am if and only if k m (mod n).
Furthermore, ||=o(a).
Corollaries to Lagrange's Theorem (restated):
(a) For any a G, o(a) is a divisor of |G|.
(b) For any a G, an = e, for n = |G|.
(c) Any group of prime order is cyclic.
3.5.1. Theorem. Every subgroup of a cyclic group is cyclic.
3.5.2 Theorem. Let G cyclic group.
(a) If G is infinite, then G Z.
(b) If |G| = n, then G Zn.
3.5.3. Proposition. Let G = be a cyclic group with |G| = n.
(a) If m Z, then
(b) The element ak generates G if and only if gcd(k,n)=1.
(c) The subgroups of G are in one-to-one correspondence with the positive divisors of n.
(d) If m and k are divisors of n, then
3.5.6. Definition. Let G be a group. If there exists a positive integer N such that aN=e for all a G, then the smallest such positive integer is called the exponent of G.
3.5.7. Lemma. Let G be a group, and let a,b G be elements such that ab = ba. If the orders of a and b are relatively prime, then o(ab) = o(a)o(b).
3.5.8. Proposition. Let G be a finite abelian group.
(a) The exponent of G is equal to the order of any element of G of maximal order.
(b) The group G is cyclic if and only if its exponent is equal to its order.