Direct Proof

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 axioms, existing lemmas and theorems, without making any further assumptions.^[1] In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations in which the statement p is true. 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. Common proof rules used are modus ponens and universal instantiation.^[2]

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. For example, instead of showing directly p ⇒ q, one proves its contrapositive ~q ⇒ ~p (one assumes ~q and shows that it leads to ~p). Since p ⇒ q and ~q ⇒ ~p are equivalent by the principle of transposition (see law of excluded middle), p ⇒ q is indirectly proved. Proof methods that are not direct include proof by contradiction, including proof by infinite descent. Direct proof methods include proof by exhaustion and proof by induction.

ExamPle::::

The sum of two even integers equals an even integerEdit

Consider two even integers x and y. Since they are even, they can be written as

${\displaystyle x=2a}$

${\displaystyle y=2b}$

respectively for integers a and b. Then the sum can be written as

${\displaystyle x+y=2a+2b=2(a+b)}$

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.

Pythagoras' TheoremEdit

Observe that we have four right-angled triangles and a square packed into a large square. Each of the triangles has sides a and b and hypotenuse c. The area of a square is defined as the square of the length of its sides - in this case, (a + b)². However, the area of the large square can also be expressed as the sum of the areas of its components. In this case, that would be the sum of the areas of the four triangles and the small square in the middle.^[5]

We know that the area of the large square is equal to (a + b)².

The area of a triangle is equal to ${\displaystyle {\frac {1}{2}}ab.}$

We know that the area of the large square is also equal to the sum of the areas of the triangles, plus the area of the small square, and thus the area of the large square equals ${\displaystyle 4({\frac {1}{2}}ab)+c^{2}.}$

These are equal, and so

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

After some simplifying,

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

Removing the ab that appears on both sides gives

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

which proves Pythagoras' theorem. ∎

The square of an odd number is also oddEdit

By definition, if n is an odd integer, it can be expressed as

${\displaystyle n=2k+1}$

for some integer k. Thus

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

Since 2k²+ 2k is an integer, n² is also odd