Direct proof

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In mathematics and logic, direct evidence is a way of showing truth or lies of statements given by a direct combination of existing facts, usually axioms, lemmas and theorems that exist, without making any further assumptions. To directly prove a conditional statement of the form "If p , then q ", that is enough to consider the situation where the statement p is correct. Logical reduction is used for reasons from assumptions to conclusions. The type of logic used is almost always first-order logic, using the numbers for all and there . The common proof rules used are ponens mode and universal instantiation.

Instead, indirect evidence can begin with certain hypothetical scenarios and then proceed to eliminate the uncertainty in each of these scenarios until inescapable conclusions are compelled. For example, instead of showing directly p => q , one proves its contrapositive ~ q => ~ p (someone assumes ~ q and indicates that it is pointing to ~ p ). Since p => q and ~ q => ~ p is equivalent to the principle of transposition (see law of the middle are excluded, p => q indirectly proved.An indirect method of evidence including evidence by contradictions, including evidence by infinite offspring Direct evidence methods include evidence by exhaustion and evidence with induction.

Video Direct proof

History and etymology

Direct evidence is the simplest evidence available. The word 'proof' comes from the Latin probare, meaning "test". The earliest use of evidence is particularly prominent in the legal process. A person with authority, like nobility, is said to have honesty, which means that the evidence is by his relative authority, which goes beyond empirical testimony. In the past, mathematics and evidence were often linked to practical questions - with populations like Egyptians and Greeks showing interest in ground surveying. This causes natural curiosity with regard to geometry and trigonometry - especially triangles and rectangles. These are forms that give many questions in practical terms, so early geometric concepts are focused on these forms, for example, people like buildings and pyramids use these forms in abundance. Another very important form in the history of direct evidence is the circle, which is essential for the design of arenas and water tanks. This means that ancient geometry (and Euclidean Geometry) discusses the circle.

The earliest form of mathematics is phenomenological. For example, if one can draw a picture that makes sense, or provides a convincing description, then it meets all the criteria for something that will be described as "facts" of mathematics. Sometimes, analogical arguments occur, or even by "begging the gods". The idea that mathematical statements can be proven has not been developed, so this is the earliest form of proof concept, although it does not become concrete evidence.

The evidence as we know comes up with one specific question: "what is the proof?" Traditionally, evidence is a platform that convinces a person without a doubt that the statement is mathematically correct. Of course, one would assume that the best way to prove the truth of something like this (B) is to make comparisons with something old (A) that has proven to be true. Thus created the concept of deriving new results from the old results.

Maps Direct proof

Example

The sum of two integers equals an even integer

Pertimbangkan dua bilangan bulat bahkan x dan y . Karena mereka genap, mereka bisa ditulis sebagai

${\ displaystyle x = 2a}  ÂÂ$

${\ displaystyle y = 2b}  ÂÂ$

masing-masing untuk bilangan bulat a dan b . Maka jumlahnya dapat ditulis sebagai

${\ displaystyle x y = 2a 2b = 2 (a b) = 2p}  ÂÂ$ di mana ${\ displaystyle p = a b}  ÂÂ$ , a dan b adalah semua bilangan bulat.

This means that x Ã, y has 2 as a factor and therefore is even, so the sum of every two even integers is even.

Pythagoras Theorem

Notice that we have four right-angled and square triangles packed into big boxes. Each triangle has sides a and b and italic c . The width of the square is defined as the square of the lengths of the sides - in this case, (a b) ² . However, the large box area can also be expressed as the sum of its component areas. In this case, it is the sum of the four triangles and the small squares in the middle.

We know that the area of ​​the big box equals (a b) ² .

Luas segitiga sama dengan ${\ displaystyle {\ frac {1} {2}} ab.}  ÂÂ$

Kita tahu bahwa luas kotak besar juga sama dengan jumlah area segitiga, ditambah luas kotak kecil, dan dengan demikian luas kotak besar sama dengan ${\ displaystyle 4 ({\ frac {1} {2}} ab) c ^ {2}.}  ÂÂ$

Ini sama, dan begitu

${\ displaystyle (a b) ^ {2} = 4 (1/2ab) c ^ {2}.}  ÂÂ$

Setelah beberapa penyederhanaan,

${\ displaystyle a ^ {2} 2ab b ^ {2} = 2ab c ^ {2}.}  ÂÂ$

Menghapus ab yang muncul di kedua sisi berikan

${\ displaystyle a ^ {2} b ^ {2} = c ^ {2},}  ÂÂ$

which proves Pythagoras' theorem. ?

The square of odd number is also odd

Menurut definisi, jika n adalah bilangan bulat ganjil, itu dapat dinyatakan sebagai

${\ displaystyle n = 2k 1}  ÂÂ$

untuk beberapa bilangan bulat k . Demikian

${\ displaystyle {\ begin {aligned} n ^ {2} & amp; = (2k 1) ^ {2} \\ & amp; = (2k 1) (2k 1) \\ & amp; = 4k ^ {2} 2k 2k 1 \\ & amp; = 4k ^ {2} 4k 1 \\ & amp; = 2 (2k ^ {2} 2k) 1. \ end {aligned}}}  ÂÂ$

Karena 2 k ² 2 k adalah bilangan bulat, n ² juga aneh. ?

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Referensi

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Sumber

• Franklin, J.; A. Daoud (2011). Bukti dalam Matematika: Suatu Pengantar . Sydney: Kew Books. ISBNÂ 0-646-54509-4. Â (Bab 1).

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Tautan eksternal

• Bukti Langsung dari Larry W. Cusick's How To Write Proofs.
• Bukti Langsung dari Pengantar Patrick Keef dan David Guichard untuk Matematika Tinggi.
• Bagian Bukti Langsung Buku Bukti Richard Hammack.

Source of the article : Wikipedia