How to Use If and Only If in Mathematics

When reading about statistics and mathematics, one phrase that regularly shows up is â€Å"if and only if.† This phrase particularly appears within statements of mathematical theorems or proofs. But what, precisely, does this statement mean? What Does If and Only If Mean in Mathematics? To understand â€Å"if and only if,† we must first know what is meant by a conditional statement. A conditional statement is one that is formed from two other statements, which we will denote by P and Q. To form a conditional statement, we could say â€Å"if P then Q.† The following are examples of this kind of statement: If it is raining outside, then I take my umbrella with me on my walk.If you study hard, then you will earn an A.If n is divisible by 4, then n is divisible by 2. Converse and Conditionals Three other statements are related to any conditional statement. These are called the converse, inverse, and the contrapositive. We form these statements by changing the order of P and Q from the original conditional and inserting the word â€Å"not† for the inverse and contrapositive. We only need to consider the converse here. This statement is obtained from the original by saying â€Å"if Q then P.† Suppose we start with the conditional â€Å"if it is raining outside, then I take my umbrella with me on my walk.† The converse of this statement is â€Å"if I take my umbrella with me on my walk, then it is raining outside.† We only need to consider this example to realize that the original conditional is not logically the same as its converse. The confusion of these two statement forms is known as a converse error. One could take an umbrella on a walk even though it may not be raining outside. For another example, we consider the conditional â€Å"If a number is divisible by 4 then it is divisible by 2.† This statement is clearly true. However, this statement’s converse â€Å"If a number is divisible by 2, then it is divisible by 4† is false. We only need to look at a number such as 6. Although 2 divides this number, 4 does not. While the original statement is true, its converse is not. Biconditional This brings us to a biconditional statement, which is also known as an if and only if statement. Certain conditional statements also have converses that are true. In this case, we may form what is known as a biconditional statement. A biconditional statement has the form: †If P then Q, and if Q then P.† Since this construction is somewhat awkward, especially when P and Q are their own logical statements, we simplify the statement of a biconditional by using the phrase if and only if. Rather than say if P then Q, and if Q then P we instead say P if and only if Q. This construction eliminates some redundancy. Statistics Example For an example of the phrase â€Å"if and only if† that involves statistics, look no further than a fact concerning the sample standard deviation. The sample standard deviation of a data set is equal to zero if and only if all of the data values are identical. We break this biconditional statement into a conditional and its converse. Then we see that this statement means both of the following: If the standard deviation is zero, then all of the data values are identical.If all of the data values are identical, then the standard deviation is equal to zero. Proof of Biconditional If we are attempting to prove a biconditional, then most of the time we end up splitting it. This makes our proof have two parts. One part we prove is â€Å"if P then Q.† The other part of the proof we need is â€Å"if Q then P.† Necessary and Sufficient Conditions Biconditional statements are related to conditions that are both necessary and sufficient. Consider the statement â€Å"if today is Easter, then tomorrow is Monday.† Today being Easter is sufficient for tomorrow to be Monday, however, it is not necessary. Today could be any Sunday other than Easter, and tomorrow would still be Monday. Abbreviation The phrase â€Å"if and only if† is used commonly enough in mathematical writing that it has its own abbreviation. Sometimes the biconditional in the statement of the phrase â€Å"if and only if† is shortened to simply â€Å"iff.† Thus the statement â€Å"P if and only if Q† becomes â€Å"P iff Q.†

Steps of CART Classification and Regression Tree - 746 Words

Overview The classic CART: Classification and Regression Tree algorithm was created by Breiman. The CART method is binary recursive partitioning procedure that can be used to process both continuous and nominal attributes as targets and predictors. The binary splits are the splitting of the data represented by nodes; each node is split into two child nodes to represent the binary split on the data into two separate paths. The recursive part of CART means that any child nodes can be additionally split into more children nodes and so forth. The partitioning refers to the data being split into multiple sections along the nodes into classifications. CART Four Main Steps The CART method involves four steps: tree creation, stopping the tree creation, tree pruning, and selecting the optimal tree. Tree creation begins with a root node, which includes all the data in the training set. CART then selects the best variable to split this node into two child nodes. To find the best variable to perform this split, CART runs through all variables and their values to determine the best combination to split the node on. The process of node splitting is repeated for each created child node and continued recursively until a stopping point is reached where no further splitting can be performed. Each node within this process is assigned a predicted class based on three factors, the assumed probability of each class, a cost matrix, and the percentage of the classifier that are located in eachShow MoreRelatedCredit Evaluation Model For Banks Using Data Mining Techniques4296 Words   |  18 PagesBackground 6 2.2.1 Decision Trees 6 2.2.2 Support Vector Machine 8 2.2.3 Logistic Regression 9 2.2.4 Ensemble Methods 10 CHAPTER 3 13 METHODOLOGY AND EXPERIMENTS 13 3.1 Data Selection and Data Preprocessing 13 3.2.1 Missing Values 14 3.2.2 Data Pre-processing 14 3.3 Results 15 3.3.1 Performance Estimation 17 CHAPTER 4 20 CONCLUSION 20 REFERENCES 21 APPENDIX 22 Outputs of Models 22 â€Æ' LIST OF FIGURES Figure 1: Logistic Regression Curve 10 LIST OF TABLES Table 1: Data Set for Analysis 13 Table 2: AreaRead MoreDiffculty Level Question Questions1681 Words   |  7 Pageslearning Option 1 and 2 both Supervised Learning Supervised Learning Tree based Modelling 2 3 Decision tree  is a type of supervised learning algorithm (having a pre-defined target variable). 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FIN Assignment Free Essays

At that point, when it disperses the unifies to the shareholders (as profits), the shareholders need to pay assesses on the profits. Fundamentally, the same wage gets burdened twice. You can stay away from this by making the corporation a â€Å"S corporation†, which implies the corporation does not pay any government wage charge itself – rather the shareholders pay assess on their offer of the corporation’s benefits. We will write a custom essay sample on FIN Assignment or any similar topic only for you Order Now This is accessible just for corporations with few shareholders. A â€Å"sole partnership† does not by any stretch of the imagination exist. You likely mean sole proprietorship. This is the point at which somebody runs heir business without the security of a corporation, partnership or limited liability organization (LLC) or any possible legal entity that ensures the managers from being subject for the business. A â€Å"partnership† is a legal entity that gives the same assessment preferences as â€Å"S corporation† – the partnership does not pay any salary charge. Partnerships come in 2 essential structures: A â€Å"general partnership† is a manifestation of business where each one accomplice is subject for the other accomplice’s exercises identifying with the business. This is clearly not a cent alternative, since limited liability is one of the primary focal points of structuring a business entity. A â€Å"limited partnership† is the place the accomplices have â€Å"limited liability† like a corporation. An alternate manifestation of association is the LLC – this provides for you the same â€Å"limited liability† as a corporation or limited partnership, and provides for you the same tax reductions as a â€Å"S corporation† or partnership. 2. Owners’ liability’ (is) limited to the amount they invested in the firm. Stockholders (are not) responsible for any encumbrances of the firm; in reticular, they (cannot) be required to pay back any debts incurred by the firm. 3. A limited liability company, or LLC, is a manifestation of business association that takes into account limited liability for the owner(s). It considers an unlimited number of parts and provides for them the liability shield that they may not have generally reveled in had they framed as a straightforward association, yet at the same time keeping up the majority of the levy profits managed by an organization. As a result of these double profits, the shareholders, or â€Å"Parts† as they are known whether some piece of a LLC, fundamentally delight in the same sorts f limited liability security that a corporation offers, with not many special cases, and in the meantime, additionally revel in certain expense focal points, including, yet not limited to, pass-through tariff and association treatment by the IRS. These focal points make All’s extremely alluring for certain business dealings and wanders. How to cite FIN Assignment, Papers

Gates Of Ivory Essay Research Paper English free essay sample

Gates Of Ivory Essay, Research Paper English 109 Gates of Ivory In a universe surrounded by war, decease, and atrociousness, it sometimes seems as if there is nowhere positive for the characters in the Gates of Ivory by Margaret Drabble to turn. In the mist of these bad images Drabble juxtaposes a alone position into the universe of adult females? s reproduction and menses that has seldom been revealed in other novels. She shows that menses exposes feelings runing from release and authorization in Alix Bowen, to dishonor, disgust and sorrow in Mme. Savet Akrun. Drabble identifies similarities between adult females on both sides of the universe, and between reproduction and adult females battling the decease of the universe? s war. Yet throughout these difficult times and uncertainnesss, the adult females in the novel show their strength and power because they hold the key to maintaining mankind alive: reproduction. Alix? s outward looks greatly exemplify what a batch of the adult females in the novel want to be. They want to experience the control that Alix displays over her organic structure. Unfortunately for Alix, this feeling is non true on the interior. Alix holds a great trade of uncertainness and insecurity within herself. This feeling of insufficiency began in early childhood. Alix feels the demand to be a savior, a defender and an militant. She wants to make? of import? and? worthy? things, but she can neer populate up to her ain criterions. In childhood, when her pet tortoise dies, she is? panicky, [ and ] after a few yearss summoned up bravery to near the immobile shell. ? ( 194 ) . The idea of non being able to assist the tortoise, or salvage him, crushed Alix. She shows how vulnerable she is to failure, inquiry and uncertainty. If everything is non perfect, Alix? s sense of control in a state of affairs disappears. The same feelings possess Alix in her grownup life. She invariably strives for the improvement of something, peculiarly adult females? s catamenial protection, but s he feels confused and scared when she becomes even the slightest spot diffident about it. She says, ? I used to postulate that if the tampon had been invented a few thousand old ages earlier, the whole history of muliebrity would hold been different. Now I? m non so certain? ( 156 ) . Alix attests that adult females? s rights, like tampons, are on? the onward March of advancement? and are? the liberator of womankind? , but what this advancement and release will convey mystifiers Alix ( 155 ; 156 ) . The fact that there is no program on precisely how to work out the job scares Alix. Alix needs concrete programs in order to be unafraid with the thoughts, Otherwise there is no warrant of success, which terrifies Alix. But even with all the insecurities, others view Alix as a strong, liberated, independent adult female. On the interior she lacks the confidence to be self-asserting as she is on the exterior. She speaks with strong words, but does non cognize how to transport out the high outlooks she and others have for her. Drabble utilizations Alix to represent both the capable, independent side, and the unsure, dependent side of a adult female. By making this, she highlights both the power and the exposure in adult females. For much of the novel Drabble hides Liz? s womanly side from the reader. It is non until in the bosom of Southeast Asia and the centre of pandemonium that her menses reappears and sends her on a whole new journey. At first glimpse she is? appalled? and it is a? daze? ( 377 ) . This attests to her surprise and incredulity, but throughout the state of affairs her saneness remains integral. While on her long journey in hunt of female protection, Liz searches through her bag and brushs? Some safety pins, a paper cartridge holder, some wooden toothpicks. A cheque book and a sea shell. An unidentifiable white pill and a gum elastic set. [ And ] Some spots of fluff? ( 378 ) . This long list of things, among others, shows how willing and unfastened Liz allows others into her life. Liz? s journey through her bag juxtaposes her journey through Southeast Asia. There are so many things that could deflect Liz on her journey, but Liz remains focused on the hunt for Stephen. When she discovers her period, distractions loom everyplace, yet protection remains in the head of her head. She encounters a whirlwind of emotions and hemorrhage controls her ideas, she can make nil else. Liz focuses her attempts to one cause and they remain at that place. This is a strength Liz uses to keep her saneness in such helter-skelter conditions. Even with the most improbable of things happening, Liz keeps her saneness and control integral and perseveres through the state of affairss. The state of affairs does non do her paranoiac, it does non even seem to phase her. Drabble utilizations Liz to demo the capacity of adult females and their finding through seeking times. Even with the biggest of distractions, adult females remain focussed and capable of anything. Unlike Alix Bowen or Liz Headeland, Mme. Savet Akrun feels non merely plume about her menses, but besides shame and sorrow. Both she and the adult females in her small town have had really dramatic experiences with their rhythms, and the poorness and the force merely aid to stress the jobs that they have had. The adult females in her small town have had to go really in melody with their organic structures and their rhythms because of the jobs they have with happening protection. Unfortunately, as Mme. Akrun says, this has led many adult females to? take to discontinue to flow? ( 153 ) . Drabble shows that it is a sense of depression, the adult females can non expect the actions of their organic structures, when they bleed, they can non protect themselves, so they will themselves into non shed blooding. Drabble illustrates that Mme. Akrun and the adult females of the small town position menses as something that can non be taken by the wars and the force of the universe around them. It must be protected, even if that means halting it for a period of clip. Yet when the menses does get down up once more, the adult females haste and reproduce every bit rapidly as possible, seeking to battle the loss of lives in the war. The protection that these adult females have for their generative system shows when Drabble uses a quotation mark that a nurse gave while in the small town: ? Every nurse? s fright was B eing taken captive and non holding any Tampax. You couldn? t count on being in the jungle and utilizing a foliage, because the jungle was defoliated. We were ever told to hold a bag packed, if we got overrun, we? d be lifted out. That was a carbon black of *censored* . I found out subsequently that they neer had any such emptying program. If we got overrun, it was merely tough breasts. In any instance, we all packed the same things. We packed money, a camera and we packed Tampax. My flax jacket was so full of Tampax that nil could hold penetrated it. ? Nurse, quoted in Nam, Mark Baker ( 159 ) In this transition Drabble shows precisely how of import menses is to adult females. Womans chose to convey Tampax over any other ownerships. Attack and invasion may take their lives, yet all they can believe about is what they will make to protect themselves when their following rhythm comes along. The nurse utilizations harsh and straightforward words. She does non keep back her feelings. She has seen the horrors and the play and the desperation in the small town, and she will non sugarcoat coat it. The same kinds of feelings possess Mme. Akrun. She has seen it all, and she refuses to dissemble it and sugarcoat coat it for others. She lives world to the fullest sense. Like the nurse, her words are non equivocal, they are certain. Both adult females realize they can merely number on themselves, and this gives a sense of release every bit good as resentment. The nurse in the transition uses phrases like? tough breasts? and? carbon black of *censored* ? to state the universe that she knows that life should non hold to be this manner, but it is and she has come to footings with it. It is all about endurance to her now. Although adult females wish they could be more protected and sheltered, they besides realize the strength and power they possess with this control, and do non desire to blow it. Even if all else fails, adult females hold the power to do a restart, a metempsychosis. It is particularly illustrated in the line? nil could hold penetrated it. ? Nothing can perforate the control these adult females have over their reproduction systems. Even through all the shame and the uncertainness, they feel they still possess control. It is the one thing in their lives that can be counted on. If it leaves, finally it will return, and no male can hold the power that they have in this country. ? The midst dark metallic urarthritiss of blood? will be the savior ( 153 ) . The enunciation that Drabble utilizations here attests to the strength of menses. It is impenetrable : it is strong and will non melt. The most of import thing that Drabble shows is the doggedness of adult females. She shows that even in the roughest times the adult females will last and maintain control. Unlike many of the adult females in the novel, Hattie Osborn sees her generative system in both a sexual and a functional manner. She views menses with a lighter attack. Hattie Osborn is most likely the most sexually provocative adult female in the novel. She openly uses her gender as a arm and a plaything. Her menses and generative system are all a gag and a game for her until Aaron Headeland comes along. Hattie alterations when this occurs, she goes from a carefree, sexually explosive being into a lovingness and concerned adult female with kid. Drabble reveals her in the beginning as holding no declinations about non holding kids: she has wholly put aside that portion of her. Then subsequently in the novel when she finds out she could be pregnant she is non in the least bit disquieted or concerned in a bad manner, she is happy. She can experience the alterations inside her? she quickens, she feels herself quicken, as the little cells bunch? and she is wholly in melody with the moti ons in her organic structure. Drabble creates a sense of repose and control in Hattie even though it is a really unsure minute. She is aroused and renewed. She is non scared of the reaction of her friend Liz, she is certain of herself. She has a sense of security that everything will turn out good somehow. Hattie had ever been the independent one, the 1 with no demand for any one else, and now all of a sudden she feels a great fond regard to Aaron. But even with this great fond regard she realizes that what is approximately to go on is particular and she can make it herself with no 1? s aid. She says? When Aaron turns to dust and ashes in my sight, and I in his, there? ll be a babe? ( 319 ) . This shows that Hattie does recognize the power she has as a adult female. She sees that there is no turning back and that what happens to her babe has nil to make with Aaron, it is all her determination now. She repeats the quotation mark? Eggs, blood and the Moon? twice in the novel ( 158, 41 6 ) . She understands the rhythm of her menses and the rhythm of her life. She is the 1 that is able to truly come to footings with her generative rhythm. She has lived the life of a free willed adult female, and now she is sing the other side of muliebrity. Drabble shows Hattie as one of the most dynamic adult females in the book. She is the lone adult female that goes through all the rhythms of muliebrity during the novel, she is the efficaciously shows the versatility and changes a adult female can travel through, and most surprisingly she stays strong and in control throughout. Drabble exemplifies Hattie? s confidence to be what all adult females should hold, no uncertainty in themselves. Drabble? s portraiture of adult females in the novel illustrates her ideas on adult females good. She shows that a adult female? s life is full of turns and bends and uncertainnesss. Throughout all the bends that each adult female had in so novel, they each came out with their caputs held high and strong. Drabble portrayed them as vulnerable and dependent in many countries, in fact in most countries. The lone portion of life a adult female had a existent clasp on was menses and reproduction, and even this was unsure at parts. Drabble seemed to be directing the message that although menses and reproduction were really independent and commanding on their ain, that adult females were the lone 1s in society that had a opportunity to command it, which gave them a batch of power. Womans hold the key to the endurance of world, and that is a great power and duty. Drabble efficaciously shows in this fresh how difficult that committedness can be.