งานม.403

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งานม.403

  1. 1. กลุ่มที่ 1 เจตน์ & ดลยา Probability In the real world events can not be predicted with total certainty. TheHow Likelybest we can do is say how likely they are to happen, using the idea ofprobability. When a coin is tossed, there are two possible outcomes: Tossing a Coin • heads (H) or • tails (T) We say that the probability of the coin landing H is 1/2. Similarly, the probability of the coin landing T is 1/2. When a single die is thrown, there are six possible Throwing Dice outcomes: 1, 2, 3, 4, 5, 6. The probability of throwing any one of these numbers is 1/6. In general:Probability
  2. 2. กลุ่มที่ 2 ตวงสิ ทธิ์ & เพญผกา ็ Probability Tree Diagrams Calculating probabilities can be hard, sometimes you add them,sometimes you multiply them, and often it is hard to figure out what to do... tree diagrams to the rescue!
  3. 3. กลุ่มที่ 3 ยุทธนา & โซลีThere are basically two types of permutation: Permutations 1. Repetition is Allowed: such as the lock above. It could be "333". 2. No Repetition: for example the first three people in a running race. You cant be first and second.
  4. 4. กลุ่มที่ 4 วรยุทธ & หัทยา ี
  5. 5. กลุ่มที่ 5 สุ ทธินันท์ & ศิวรักษ์ CombinationsThere are also two types of combinations (remember the order does not matter now): 1. Repetition is Allowed: such as coins in your pocket (5,5,5,10,10) 2. No Repetition: such as lottery numbers (2,14,15,27,30,33)Actually, these are the hardest to explain, so I will come back to this later.1. Combinations with RepetitionThis is how lotteries work. The numbers are drawn one at a time, and if you have the2. Combinations without Repetitionlucky numbers (no matter what order) you win!The easiest way to explain it is to: • assume that the order does matter (ie permutations), • then alter it so the order does not matter.Going back to our pool ball example, let us say that you just want to know which 3 poolballs were chosen, not the order.We already know that 3 out of 16 gave us 3,360 permutations.But many of those will be the same to us now, because we dont care what order!For example, let us say balls 1, 2 and 3 were chosen. These are the possibilites:
  6. 6. กลุ่มที่ 6 เอกรัฐ & อญธิกา ัExponent Properties
  7. 7. กลุ่มที่ 7 พินิจ & สมฤทัย
  8. 8. กลุ่มที่ 8 พีระพงษ์ & ปนัสยา
  9. 9. กลุ่มที่ 9 วนดี & ศิริรัตน์ ั Complex Numbers A complex number is made up of both real and imaginarycomponents. It can be represented by an expression of the form (a+bi),where a and b are real numbers and i is imaginary. When defining i we saythat i = . Then we can think of i2 as -1. In general, if c is any positivenumber, we would write: . If we have a complex number z, where z=a+bi then a would be thereal component (denoted: Re z) and b would represent the imaginarycomponent of z (denoted Im z). Thus the real component of z=4+3i is 4 andthe imaginary component would be 3. From this, it is obvious that twocomplex numbers (a+bi) and (c+di) are equal if and only if a=c and b=d,that is, the real and imaginary components are equal. The complex number (a+bi) can also be represented by the orderedpair (a,b) and plotted on a special plane called the complex plane or theArgand Plane. On the Argand Plane the horizontal axis is called the real axisand the vertical axis is called the imaginary axis.
  10. 10. กลุ่มที่ 10 ออมสิ น & ชฎาพร Properties of the Complex SetThe set of complex numbers is denoted . Just like any other number set there arerules of operation.The sum and difference of complex numbers is defined by adding or subtracting theirreal components ie: The communitive and distributive properties hold for the product of complexnumbers ie: When dividing two complex numbers you are basically rationalizing thedenominator of a rational expression. If we have a complex number defined as z =a+bithen the conjuate would be . See the following example:Example:
  11. 11. กลุ่มที่ 11 ชวลวทย์ & รัตนาภรณ์ ั ิ Conjugates The geometric inperpretation of a complex conjugate is thereflection along the real axis. This can be seen in the figure below where z =a+bi is a complex number. Listed below are also several properties ofconjugates.Properties:
  12. 12. กลุ่มที่ 12 วศิน & ผกามาศ Absolue Value/Modulus The distance from the origin to any complex number is the absolutevalue or modulus. Looking at the figure below we can see that PythagorasTheorem gives us a formula to calculate the absolute value of a complexnumber z = a+bi And from this we get:There are also some properties of absolute values dealing with complexnumbers. These are:
  13. 13. กลุ่มที่ 13 กาญจนา & สุ กญญา ัSet Notation
  14. 14. กลุ่มที่ 14 เกศินี & สุทธิมาสSet
  15. 15. กลุ่มที่ 15 ธารารัตน์ & ชัญญานุชElements of Set
  16. 16. กลุ่มที่ 16 สุ ภาพร & ศิริพรVenn Diagrams
  17. 17. กลุ่มที่ 17 อริสา & อรสาComplement of a Set
  18. 18. กลุ่มที่ 18 วราภรณ์ & วจีพรUnion of Sets
  19. 19. กลุ่มที่ 19 พรธิชา & วรอนงค์The Intersection of Sets
  20. 20. กลุ่มที่ 20 โกสิทธ์ิ Difference of setsThe difference of sets “A-B” is the set of all elements of “A”, which do not belong to“B”.In the set builder form, the difference set is : A − B = {x x ∈ A ∧ x ∉ B}or A − B = A ∩ BThe difference of sets “B - A” is the set of all elements of “B”, which do not belongTo “A”.In the set builder form, the difference set is : B − A = {x x ∈ B ∧ x ∉ A}or B − A = B ∩ A

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