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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. 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.
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.
What follows is a simple, direct proof that the sum of two even integers is itself an even number.
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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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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The discography of Soulja Boy, an American recording artist and producer, consists of four studio albums, one extended play (EP), thirty-nine mixtapes, ten singles, thirty-eight music videos.
After releasing his first mixtape, Unsigned & Still Major: Da Album Before da Album, Soulja Boy released his major-label debut souljaboytellem.com in 2007; which included the hit single "Crank That (Soulja Boy)". The same year, he was granted his own imprint label Stacks on Deck Entertainment. In 2008, he released his second studio album, iSouljaBoyTellem; the album produced the top ten Billboard Hot 100 single "Kiss Me Thru The Phone". In 2010, he released his third studio album, The DeAndre Way; which included the singles "Pretty Boy Swag", "Blowing Me Kisses", and "Speakers Going Hammer".
In 2015, he released two studio albums titled Loyalty and Blessed under the label SOD.
Note: The singles appeared on either the Bubbling Under Hot 100 or Bubbling Under Hot R&B/Hip-Hop Songs, 25 song extensions of the Billboard Hot 100 and Hot R&B/Hip-Hop Songs, respectively.
Live and direct
Live and direct [free scratch]
We can go out
Love what we be about [x2]
We can go out
Love what we be about [x2]
We can go out on the weekend
Cause the weekend is free
Stayin Good
We can off up the deep end if ya thinking of me
We can make it on your old mans money
"da dup" "da dup
Never make it on mine
"da dup" "da dup")(KRS-ONE "szzump da
dump da dump"
We can make it through the weekend
"da dup" "da dup
You believe in me
"da dup" "da dup
Your the one that said to me
"ya ya ya ya ya"
Your the one that said to me
That we dont regret anymore
"whata bye bye bye whata bye bye
bye whata bye bye bye"
We dont regret anymore
"Love what we be about" [x2]
We can get out on the weekend everybody can see
We can drive off the deep end if you depend on me
Live
direct
We can make it
"da dup" "da dup"
You can make it
"da dup" "da dup"
Never make it on time
"da dup" "da dup" [KRS-ONE] "szzump da
dump da dump"
Loose your mind on the weekend
"da dup" "da dup"
Off the deep end with me
"da dup" "da dup" "da dup" "da dup"
Hey sit down put your money on the ground cause its gonna be a tragedy
"woulda bye bye bye
Hey sit down put your money on the ground cause its gonna be a tragedy
"ya ya ya ya ya
Your the one that said to me
Your the one that said to me
That we dont regret anymore
"ya ya ya ya ya, whata bye bye bye whata bye bye bye whata
bye bye bye"
We dont regret anymore
"Love what we be about" [x2]
Get it up get it up
Get it up get it up
"get it up get it up"