The importance of math


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The importance of math

  1. 1. Course sets 7 Class mathematics Subject Submitted by Sayeda Tayyaba Shahid (1409) Tatheer Zahra (1405) Saba Shaukat (1442) Submitted to Shafiq ur rehman
  2. 2. The Importance of Math Why is Math so important? Probably because it is used in so many other subjects. There are uses of mathematics in all the "hard" sciences, such as biology, chemistry, and physics; the "soft" sciences, such as economics, psychology, and sociology; engineering fields, such as civil, mechanical, and industrial engineering; and technological fields such as computers, rockets, and communications. There are even uses in the arts, such as sculpture, drawing, and music. In addition, anything which uses a computer uses mathematics, and you probably are aware of how many things that is! Furthermore, learning mathematics forces one to learn how to think very logically and to solve problems using that skill. It also teaches one to be precise in thoughts and words. Math teaches life skills. It is difficult to find any area of life that isn't touched by mathematics. We are surrounded by math, and also surrounded by people who do know math. If you don't know what's going on, you are at their mercy. Through which ICT we share our objective      By using internet social media E-mail Multimedia computer
  3. 3. Objectives  Express a set in  Descriptive way  Set builder form  Tabular form  Define intersection, union and different of two sets.  Find  Union of two or more sets  Intersection of two or more sets  Difference of two sets     Define and identify disjoint and overlapping sets. Define universal set and complement of a set. Represent set through venn diagram. Perform operations of union, intersection, difference and complement on two sets A and B, when:     A is subset of B, B is subset of A, A and B are disjoint sets, A and B are overlapping sets, through venn diagram.
  4. 4. Set theory Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. What is set? A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, the founder of set theory, gave the following definition of a set. 1. A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought – which are called elements of the set. 2. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.
  5. 5. Describing sets There are two ways of describing, or specifying the members of, a set. One way is by intentional definition, using a rule or semantic description: A is the set whose members are the first four positive integers. B is the set of colors of the French flag. The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets: C = {4, 2, 1, 3} D = {blue, white, red}. Every element of a set must be unique; no two members may be identical. (A multiset is a generalized concept of a set that relaxes this criterion.) All set operations preserve this property. The order in which the elements of a set or multiset are listed is irrelevant (unlike for a sequence or tuple). Combining these two ideas into an example {6, 11} = {11, 6} = {11, 6, 6, 11} because the extensional specification means merely that each of the elements listed is a member of the set. Membership The key relation between sets is membership – when one set is an element of another. If a is a member of B, this is denoted a ∈ B, while if c is not a member of B then c ∉ B. For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = {n2 – 4 | n is an integer; and 0 ≤ n ≤ 19} defined above, 4 ∈ A and 12 ∈ F; but 9 ∉ F and green ∉ B. Expressing a set There are three ways to express a set 1) Descriptive form 2) Tabular form 3) Set builder form
  6. 6. 1) Descriptive form: IF a set is described with the help of a statement, it is called descriptive form . for example N= set of natural numbers Z= set of integers P= set of prime numbers W=set of whole numbers Natural Number The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... or to the set of nonnegative integers 0, 1, 2, 3, ... Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers. In fact, Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P." The set of natural numbers (whichever definition is adopted) is denoted N. Due to lack of standard terminology, the following terms and notations are recommended in preference to "counting number," "natural number," and "whole number." Set Name .., -2, -1, 0, 1, 2, ... 1, 2, 3, 4, ... 0, 1, 2, 3, 4, ... Integers positive integers Non-negative integers Non-positive integers negative integers 0, -1, -2, -3, -4, ... -1, -2, -3, -4, ... symbol Z Z-+ Z-* Z-- Integers An integer is a number that can be written without a fractional or decimal component. For example, 21, 4, and −2048 are integers;
  7. 7. 9.75, 5½, and √2 are not integers. The set of integers is a subset of the real numbers, and consists of the natural numbers (1, 2, 3, ...), zero (0) and the negatives of the natural numbers (−1, −2, −3, ...). Prime number A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because only 1 and 5 evenly divide it, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6 Whole number Whole numbers may variously refer to:  natural numbers beginning 1, 2, 3, ...; the positive integers  natural numbers beginning 0, 1, 2, 3, ...; the non-negative integers  all integers ..., -3, -2, -1, 0, 1, 2, 3, ... 2) Tabular form: If we list all elements of a set with in the braces {} and separate each elements by using a comma , it is called the tabular form . For example A={a,e,i,o,u} C={3,6,9…..99} W={0,1,2,3…..}
  8. 8. 3) Set builder form: If a set is described by using a common property of all its element, it is called set builder form. For example "E is a set of even of even numbers" Some important symbols ∧ ≤ ≥ ∈ V | Such that Less than or equal to Greater than and equal to Belongs to or Such that Mathematics 7 by Sh. M. Tariq Rafiq, Zulqarnain Ansari
  9. 9. Exercise  Write the following sets in descriptive form. 1) A= {a,e,i,o,u} A= vowels of English alphabets. 2) B= {6,7,8,9,10} B= set of natural numbers from 6 to 10.  Write the following sets in tabular form. 1) E= Letters of the word “hockey”. E= {h,o,c,k,e,y} 2) T= Multiples of 5 less than 30. T= {5,10,15,20,25}  Write the following sets in set builder form. 1) Z= set of natural numbers. Z= {x | x ∈ N} 2) O= set of odd number greater than 15. O={x | x ∈ O ∧ x >15}
  10. 10. Operations on sets  Union, Intersection and difference of two sets  Union of three sets Following are the steps to find union of three sets Step 1 Find the union of any two sets. Step 2 Find the union of remaining 3rd set and the set that we get as the result of the first step. For three sets A, B and C there union can be taken in any of the following way. A ∪ (B∪C) and ={1,2,3,4} ∪ [{3,4,5,6,7,8} ∪ {6,7,8,9,10}] ={1,2,3,4} ∪ {3,4,5,6,7,8,9,10} = Intersection of three sets A ∩ (B∩C) , and = {a, b, c, d} ∩ ({c, d, e} ∩ {c, e, f, g}) = {a, b, c, d} ∩ {c, e} = Difference of two sets Difference of set means (A-B) = {1, 3, 6,} - {1, 2, 3, 4, 5} =
  11. 11. Exercise  Find the union of the following sets  A∪B=? , = {1, 3, 5} ∪ {1, 2, 3, 4} = {1, 2, 3, 4, 5}  Find the intersection of the following sets , = {3, 6, 9, 12, 15} ∩ {5, 10, 15, 20} = {15} Disjoint and overlapping Sets  Disjoint Sets Two sets A and B are said to be disjoint sets, if there is no common element between them. i.e. A = {1, 2, 3} and B = {4, 5, 6} are disjoint sets because there is no common element in set A and set B.  Overlapping sets Two sets A and B are called overlapping sets , if there is at least one element common between them but none of them is a subset of other i.e. A = {0 , 5, 10} and B = {1, 3, 5, 7} are overlapping sets because 5 is a common element in each sets A and B.
  12. 12. Universal Set and Complement of a set  Universal Set A set which contains all the possible elements of the sets under consideration is called the universal set. For example, the universal set of the counting numbers means a set that contains all possible numbers that we can use for counting. To represent such a set we use the symbol ∪ and read it as "universal se".  Complement of a set Consider a set B whose universal set is U then the difference set UB or ͨͨͨͨͨͨ ͨͨͨ U-B is called the complement of set B , which is donated by B' or B and read as "B complement ". If U = {1, 2, 3... 10} and B = {1, 3, 7, 9} then find the B' = {1, 2, 3…..10} - {1, 3, 7, 9} = {2, 4, 5, 6, 8, 10} Venn diagram A venn diagram is simply closed figure to show sets and the relationship between difference sets.
  13. 13. Mathematics 7 by Sh. M. Tariq Rafiq; Zulqarnain Ansari
  14. 14.    Mathematics 7 by Sh. M. Tariq Rafiq; Zulqarnain Ansari