In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Logic is the study of the principles of valid demonstration and Inference. The meaning of the word truth extends from Honesty, Good faith, and Sincerity in general to agreement with Fact or Reality In Mathematics, a lemma (plural lemmata or lemmas from the Greek λήμμα "lemma" meaning "anything which is received In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements In order to directly prove a conditional statement of the form "If p, then q", it is only necessary to consider situations where the statement p is true. The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain Conditionals in Logic Logical deduction is employed to reason from assumptions to conclusion. The type of logic employed is almost invariably first-order logic, employing the quantifiers for all and there exists. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science Common proof rules used are modus ponens and universal instantiation. In Classical logic, modus ponendo ponens ( Latin: mode that affirms by affirming; often abbreviated to MP or modus ponens) is a In Logic universal instantiation ( UI, sometimes confused with Dictum de omni) is an Inference from a truth about each member of a class of individuals
In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. In Classical logic, modus tollens (or modus tollendo tollens) ( Latin for "the way that denies by denying" has the following Argument form For example instead of showing directly p → q, one proves its contrapositive ~q → ~p (one assumes ~q and shows that it leads to ~p). For contraposition in the field of traditional logic see Contraposition (traditional logic. Since p → q and ~q → ~p are equivalent by the principle of transposition, one has indirectly proved p → q. Transposition (mathematics -->In the methods of Deductive reasoning in Classical logic, " Proof methods that are not direct include Proof by contradiction, Proof by exhaustion, Proof by infinite descent and Proof by induction. Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of Mathematical proof in which In Mathematics, a proof by infinite descent is a particular kind of proof by Mathematical induction. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that
Example
What follows is a simple, direct proof that the sum of two even integers is itself an even number. In Mathematics, the parity of an object states whether it is even or odd
Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clear x + y has 2 as a factor and therefore is even, so the sum of any two even integers is even.
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