Mathematics in the Modern World, Summaries of Mathematics

Download Mathematics in the Modern World and more Summaries Mathematics in PDF only on Docsity! LANGUAGE OF MATHEMATICS An expression is a mathematical statement that does not contain an equal sign. It cannot be solved for unless the value of the variable is given. Subtraction (-) minus A number minus seven X-7 less than : Four less than a number a x-4 the difference of | The difference of a » number an and three _ x-3 less - Nine less anumber 9-x decreased by | Anumber decreased by twelve x=12 subtracted from - Six subtracted from a number x-6  Multiplication (x) | times the product of twice; double multiplied by Eight times a number A number multiplied by negative six 8x 14x The product of fourteen and a number Twice a number, double a number - Three fourths of a number  Division (+) the quotient of The quotient of a number and seven divided by Ten divided by a number x |S |~lx the ratio of The ratio of a number to fifteen **Special note: When you encounter one of these three phrases, you need to invert (switch the order) your translation. “Read the written expression left to right. Translate written words into mathematical operators. Look out for those SPECIAL phrases that make us switch things around!” ** less than ** fewer than ** subtracted from Four less than a number is 10. x—4=10 The product of 10 and a number is greater than 260. 10y > 260 The sum of K and 9, divided by 10 (K+ 9)=+ 10 M more than the product of X and 5 M + (X x 5) R divided by T, minus 6 (R=+T)-—6 The sum of A and 9 A+9 The quotient of 7 and K 77K BASIC MATHEMATICAL CONCEPT Examples: A= fa, e,1, 0, u} B = {set of plane figures} C = {Ca, Au, Ag} A set can be represented by listing its elements between braces: A = (1,2,3,4,5,6,7,8,9,0}. This is the tabular or roster form. The symbol € is used to express that an element is part of a set (or belongs to a set), for instance 3 € A. An alternative way to define a set, called set builder notation, is by stating a property (predicate) P(x) verified by exactly its elements, for instance A={xEZ|1sxs 5} = “set of integers x such that 1 < x < 5’—i.e.: A = {1,2,3,4,5}!. In general: A = {x € U | p(x)}, where U is the domain of discourse in which the predicate P(x) must be interpreted, or A = {x | P(x)! if the domain of discourse for P(x) is implicitly understood. {x]...!. is read as “x such that. In set theory the term universal set is often used in place of “domain of discourse” for a given predicate. Others call it as the rule form. Illustrations: Roster Rule 41,2,3} {x | x is a natural number less than 4} {-.=,2,- = {0,=,=,5} | is a whole number less than 4} {2,4,6,8,10} {x | x is an even integer between O and 12} {0,3,6,9} {k | kis a multiple of 3 between -1 and 12} Some important sets are the following: 1. N = {1,2,3,--} = the set of natural numbers. 2. W = {0,1,2,3,--} = the set of whole numbers. 3. Z = {-3,-2,-1,0,1,2,3,-} = the set of integers. 4. Q = the set of rational numbers (terminating or repeating decimals). 5. Q’ = the set of irrational numbers (nonterminating, nonrepeating decimals). 6. R =the set of real numbers. 7. C =the set of complex numbers. Set Ais equal to set B, denoted by A = B, if and only if A and B have exactly the same elements. Example: A= {h,o, p.e} B=({p,o,e,h} Set Ais equivalent to set B, denoted by A ~ B, if and only if Aand B have the same number of elements. The cardinality of the two sets is the same. Example: A=l,0,v,e} B=\a. 8.9.0} The sets C = {a, b, c} and D = {4, 5, 6} are equivalent sets. Also, {2} ~ 1}. Equal sets are equivalent, but not vice versa. Sets that have common elements are joint sets. The sets A = {4, 5, 6} and B = {6, 10, 11} are joint sets, since 6 is a common to both A and B. The sets C = {r, I, c, h} and D = fp, o, b, r, e} are joint sets because r is common to both C and D. Two sets are disjoint if they have no common elements. The set E = {a , b, c} and F = {e, f, g} are disjoint sets, since no element is common. The set {0} and {@} are also disjoint sets. The positive odd integer Zo = {1, 3, 5, ...| and the nonnegative even integers Z. = {0, 2, 4, ...} are disjoint sets. Also, the negative integers Z = {-1, -2, ...} and the nonnegative integers W = {0, 1, 2, ...} are disjoint sets. Set A is a subset of set B denoted by A cB, if every element of A belongs to B. In Symbol, Ac Bifx € A, then x € B. Aside from the definition, if there is at least one element found in B but not in A, then A is a proper subset of B denoted by Ac B. There are two improper subsets of any given set, the empty set and the set itself. The power set P of A, denoted by P(A) is defined as the set of all subsets of A. The following generalizations are consequences of the definition. a. Every set is a subset of itself, i.e. AC A. b. An empty set is always a subset of every set, i.e. @ C A. c. The sets {@} and {0} are not empty, since each contains one element. The English logician John Venn (1834-1923) developed diagrams that can be used to illustrate sets and relationships between sets. This diagram facilitated one’s conceptualization of the sets and relations within it. It is called the Venn Diagram. In a Venn diagram, the universal set is represented by a rectangular region and subsets of the universal set are generally represented by oval or circular regions drawn inside the rectangle. Others would prefer different types of polygons to emphasize differences between them. The Venn diagram below shows a universal set and one of its subsets, labeled as set A. The size of the circle is not a concern. The region outside of the circle, but inside of the rectangle, represents the set A’. u A Venn Diaaram Four Basic Operations on Sets: 1. Union of Sets A and B A U B & sets of all elements found in A or B or both = & | x €Aorx € B} Example: fa, b,c, d, e} U {b, e, f, g} = fa, b,c, d, e, f, g} In general: AUU=U,AU@=A,AUA=A 2. Intersection of Sets A and B Af B = set of all elements common to both A and B = {x | x € Aand xé€B Example: {1, 2, 3, 441 {0, 2, 3, 4, 9} = {2, 3, 4} In general: ANU=A,AN@=G6,ANA=A Laws of Sets Sets involving the operations union, intersection, complement and difference satisfy properties which we shall refer to as the laws of sets. 1. Commutative Law — The order in which the sets are taken does not affect the result. AUB=BUA ANB=BNA Examples: {2} U {3} = {3} U {2}; {2} {3} = {3}. {2} 2. Associative Laws — The grouping in which the sets are taken does not affect the result. AU(BUC)=(AUB)UC AN(BNC)=(ANBNC Examples: [{a} U fb, oj] U {c, e, f= fa} U [fb, c} U {c, €, 8] 3. Identity Laws - A set operated to another set called the identity element gives the set itself. AU @ =A, for union of sets, the identity is the empty set. AU =A, for intersection of sets, the identity element is the universal set. 4. Inverse or Complement Laws - This involves inside and outside of a set. AUA’=U ANAN=6 5. Distributive Laws - These laws involve three sets with two different operations, distributing the first operation over the second one. AU(BNC)=(AUB)N (AUC); Left Distributive Law of U over 0. AN (BUC)=(ANB)U(ANC); Left Distributive Law of 0 over u. (AN B)UC=(AUC)N (BUC); Right Distributive Law of U over N. (AUB)NC=(ANC)U (BNC); Right Distributive Law of N over vu. B. RELATIONS AND FUNCTIONS Relations abound in daily life: people are related to each other in many ways as parents and children, teachers and students, employers and employees, and many others. In business things that are bought are related to their cost and the amount paid is related to the number of things bought. We also look at the relation of the prices as the supply is increased or reduced. In geometry, we say that the area is also related to the volume. In physics distance travelled is related to the velocity. In general, we relate one set of information to another. Thus, any correspondence between the elements of two sets is a relation. Mathematically; a relation is a correspondence between two things or quantities. It is a set of ordered pairs such that the set of all first coordinates of the ordered pairs is called Domain and the set of all the second coordinates of the ordered pairs is called Range. A relation maybe expressed as a statement, by arrow diagram, through table, by an equation, or graphically. Evaluating a Function The functional notation y = f(x) allows us to denote specific values of a function. To evaluate a function is to substitute the specified values of the independent variable in the formula and simplify. Example: When f(x) = 2x — 3, find (2) Solution: f(2) = 2(2)-3=4-3 i(2) = 1 a. If f(x) = 2x2- 3x +5, find a. f(4) b. f(-3) c. c. f(5) Solution: a. {(4) = 2 (4)2- 3(4) + 5 = 2(16)- 12+ 5 =32-12+5=25 b. f(-3) = 2 (-3)2 - 3(-3) +5 = 299) +9+5=18+94+5=32 c. £(5) = 2 (5)2- 3(5) + 5 = 2(25)- 15+5=50-15+5-=40 Fundamental Operations on Functions 1. The sum/difference of two functions f and g is the functions defined by (f + g)(x) = f{x) + gfx) The resultant function is the algebraic sum of the two functions. 2. The product of two functions fand g is the function defined by (fg)(x) = f(x) g(x) The resultant function is equal to the product of the separate images. 3. The quotient of two functions f and g is defined by the function (5) 9) =, ax) <0 The resultant function is equal to the quotient of the separate images. Example 1: If f(x) = 8 - 3x and g(x) = 5-x, find a. (Fg)x) —b. (Fg)(2)-ffgyis) ed. (EY) Solutions: a. (f+g)(x) = f(x) + g(x) = (8 - 3x) + (5-x) =8+5-3x-x=13-4x b. (f- g)x) = f(x) - g(x) = (8 - 3x) -(5 -x) = 8-5-3x-(-x) =3 - 2x c. (fg)(x) = flx)g(x) = (8 - 3x) (5 — x) = 40 - 23x + 3x2 d. -) (x) = 8-3x