Mathematics-in-the-Modern-world-Mathematical-Language-and-Symbols.pdf

Mathematical Language and Symbols Mathematics as a language language Consists of words used in a structured and conventional way and conveyed by speech, writing, or gesture. Mathematics as a language Expression – a group of number or variable with or without mathematical operation...

Mathematical Language and Symbols Mathematics as a language language Consists of words used in a structured and conventional way and conveyed by speech, writing, or gesture. Mathematics as a language Expression – a group of number or variable with or without mathematical operation - sum of two numbers → x + y Equation – a group of number or variable with or without mathematical operation separated by an equal sign - the difference between two number is 10 → x – y = 9 Classify the following as expression or equation 1. The product of two numbers 2. The sum of three integers is greater than 11 3. Half of the sum of 23 and 88 4. 2x – 3 5. x=1 6. x+ 3y/2 7. x + 2x + 3x + 4x + 5x Characteristic of mathematical language Precise – able to make a fine distinctions Concise – able to say things briefly Powerful – able to express complex thoughts with relative ease Sets A collection of distinct, well-defined objects forming a group The objects inside a set is called elements Sets are named by using any capital letter Representation of sets Roster method - A method of listing the elements of a set in a row with comma separation within curly brackets is called roster notation Set – builder notation - a representation or a notation that can be used to describe a set that is defined by a logical formula that simplifies to be true for every element of the set Representation of sets Roster method Set T = { 1, 2, 3, 4, 5} Set A = {a, b, c, d, e, f} Set – builder notation Set G = { h | h ≠ 20} read as “set G is a set of real number, such that h is any real number except for 20” Set V = { m| m < 10} Read as “ set V is a set of real number m, such that m is less than 10} Types of sets Singleton sets Null or empty sets Equal sets Subsets Universal sets Disjoint sets Power sets Singleton A set that contains one element Set A = { x| x is a whole number between 15 and 17} Null or Empty set A set that does not contain any element Denoted by φ (phi) Set B = { integers between 0 and 1} Set X = { x| x is an even prime number greater than 3 but less than 11} Equal sets Two sets are equal if they have the same elements in them Equivalent sets Two sets that have the same number of elements, even if the elements are different Subsets If every element in set A is also present in set B then, set A is a subset of set B Cardinality of a finite set defined as the number of elements in a mathematical set. It can be finite or infinite. The cardinality of a finite set A is denoted by |A|, n(A) If A = {a, b, c, d, e}, then n(A) (or) |A| = 5 Cardinality of a Power Set Is the set of all subsets of the given set. If a set A has n elements, then the cardinality of its power set is equal to 𝟐𝒏 which is the number of subsets of the set A If a set A = {1, 2, 3, 4}, then the cardinality of the power set of A is 𝟐𝟒 = 16 as the set A has cardinality 4. Universal set The collection of all the elements in regard to a particular subject is known as a universal set which is denoted by the letter “U.” Disjoint set Two sets are known as disjoint sets if they have no common elements in both sets. Operations on Sets Union of Sets Intersection of Sets Difference of Sets Complement of a set Union of Sets Set of elements that are present in set A and set B , denoted by A U B Intersection of Sets Denoted by A ∩ B, means that elements which are common to set A and set B Difference of Sets We denote the difference of sets by A – B which means the elements in set A that are not in set B. Complement of a Set The complement of set A is defined as a set that contains the elements present in the universal set but not in set A Denoted as “ A’ “ Functions and Relations Ordered Pair A pair of elements or numbers written in a specific and fixed order On a Cartesian plane, an ordered pair (x, y) is defined as the coordinates of a point such that “x” is the x-coordinate and “y” is the y-coordinate. Relations a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. Domain The set of all x or input values. We may describe it as the collection of the first values in the ordered pairs. Range the set of all y or output values. We may describe it as the collection of the second values in the ordered pairs. Find the Domain and Range of the following ordered pair Functions is also a set of ordered pairs; however, every x-value must be associated to only one y-value. Relations and Functions Every function is a relation, but not all relations are functions Determining whether the set of ordered pair is Function or Relation We can check if a relation is a function either graphically or by following the steps below. Examine the x or input values. Examine also the y or output values. If all the input values are different, then the relation becomes a function, and if the values are repeated, the relation is not a function. Determine if the given set is function or relation 1. {(-2, 3), {4, 5), (6, -5), (-2, 3)} 2. W= {(1, 2), (2, 3), (3, 4), (4, 5) 3. Y = {(1, 6), (2, 5), (1, 9), (4, 3)} 4. R = (1,1); (2,2); (3,1); (4,2); (5,1); (6,7) Binary Operation a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set. A binary operation *, on the set of real numbers, is a rule which combines any two real numbers a and b and gives a real number. Binary Operation Examples include the familiar elementary arithmetic operations of addition, subtraction, multiplication and division. Other binary operations *, Ꙩ, ∆, , , Example of Binary Operation a * b =c 4+5=9 9–2=7 7 x 3 = 21 81 ÷ 3 = 27 Other Binary Operation can be defined as a * b = a + b - ab Evaluate 4 * 3 = 4 + 3 – (4. 3) = 7 – 12 = -5 By using the given binary operation above Evaluate 7 * 4 A binary operation * is defined as a * b = 4a – b Evaluate 4*5 Properties of Binary Operation Closure property Commutative property Associative property Identity element Inverse property Closure property A set is said to be closed under binary operation * if any two real numbers from the set , the result of the binary operation returns to a member of the set  Closure property A binary operation * is defined on If set P = {0, 2, 4} and x ∆ the set D = {1, 2, 3} as, a * b = 2a – y = x + y – ½ xy. Is the set b. is the set T closed under *? P closed under ∆? * 1 2 3 ∆ 0 2 4 1 0 2 2 3 4 Commutative property A binary operation * defined on a set R of real numbers is commutative if a * b = b * a for all a, b ∈ ℝ Sample problem for Commutative property Is a * b = a + b + ab commutative for all a, b ∈ ℝ? a*b=b*a a + b + ab = b + a + ba → yes, therefore * is commutative Commutative property A binary operation is defined x * y = 2x – y , is the operation commutative? Evaluate 2*3 3*2 Commutative property A binary operation * is commutative if - the elements are reflected about the leading diagonal * a b c a b a c b a c b c c b a Associative property A closed binary operation * of real numbers is associative if (a * b ) * c = a * (b * c ) for all a, b, c ∈ ℝ Take note: + and x are associative. Associative property If a * b = a + ab, Is the binary operation * associative ? Evaluate the binary operation (2 * 3) *4 and 2* (3 * 4) if the binary operation is defined as a * b = a + b + ab a) (2 * 3) * 4 = b) 2* (3 * 4) = Identity element For a binary operation *, if their exist just one element e such that e * a = a and a * e = a Where a, e ∈ R For addition: e = 0 For multiplication:e = 1 Inverse property An element 𝑎−1 is is called an inverse of a number under the binary operation * if a * 𝑎−1 = e similarly 𝑎−1 * a = e