Fully differential amplifier

2013-10-16T07:51:16Z

115.119.135.242: /* AC imperfections */

: ''For contraposition in the field of traditional logic, see [[Contraposition (traditional logic)]].''
: ''For contraposition in the field of symbolic logic, see [[Transposition (logic)]].''

In [[logic]], '''contraposition''' is a law, which says that a [[conditional statement]] is [[logically equivalent]] to its '''contrapositive'''. The contrapositive of the statement has its antecedent and consequent [[Inverse (logic)|inverted]] and [[Conversion (logic)|flipped]]: the contrapositive of <math>P \rightarrow Q</math> is thus <math> \neg Q \rightarrow \neg P </math>. For instance, the proposition "''All bats are mammals''" can be restated as the conditional "''If something is a bat, then it is a mammal''". Now, the law says that statement is identical to the contrapositive "''If something is not a mammal, then it is not a bat''."

The contrapositive can be compared with three other relationships between conditional statements:

*'''[[Inverse (logic)|Inversion]]''' (the '''inverse'''): <math>\neg P \rightarrow \neg Q</math>.
"''If something is not a bat, then it is not a mammal''." Unlike the contrapositive, the inverse's [[truth value]] is not at all dependent on whether or not the original proposition was true, as evidenced here. The inverse here is clearly not true.
*'''[[Conversion (logic)|Conversion]]''' (the '''converse'''): <math>Q \rightarrow P</math>.
"''If something is a mammal, then it is a bat''." The converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition.
*'''[[Negation]]''': <math>\neg (P \rightarrow Q)</math>.
"''There exists a bat that is not a mammal''. " If the negation is true, the original proposition (and by extension the contrapositive) is untrue. Here, of course, the negation is untrue.

Note that if <math>P \rightarrow Q</math> is true and we are given that ''Q'' is false, <math>\neg Q</math>, it can logically be concluded that ''P'' must be false, <math>\neg P</math>. This is often called the ''law of contrapositive'', or the ''[[modus tollens]]'' rule of inference.

==Intuitive explanation==

[[File:Venn A subset B.svg|thumb|right|]]

Consider the [[Euler diagram]] on the right. According to this diagram, if something is in A, it must be in B as well. So we can interpret "all of A is in B" as:

:<math>A \to B</math>

It is also clear that anything that is '''not''' within B (the white region) '''cannot''' be within A, either. This statement,

:<math>\neg B \to \neg A</math>

is the contrapositive. Therefore we can say that

:<math>(A \to B) \to (\neg B \to \neg A)</math>.

Practically speaking, this may make life much easier when trying to prove something. For example, if we want to prove that every girl in the United States (A) is blonde (B), we can either try to directly prove <math>A \to B</math> by checking all girls in the United States to see if they are all blonde. Alternatively, we can try to prove <math>\neg B \to \neg A</math> by checking all non-blonde girls to see if they are all outside the US. This means that if we find at least one non-blonde girl within the US, we will have disproved <math>\neg B \to \neg A</math>, and equivalently <math>A \to B</math>.

To conclude, for any statement where A implies B, then ''not B'' always implies ''not A''. Proving or disproving either one of these statements automatically proves or disproves the other. They are fully equivalent.

==Formal definition==

A proposition ''Q'' is implicated by a proposition ''P'' when the following relationship holds:

:<math>(P \to Q)</math>

In [[vernacular]] terms, this states that, "if ''P'', then ''Q''", or, "if ''Socrates is a man'', then ''Socrates is human''." In a conditional such as this, ''P'' is the [[Antecedent (logic)|antecedent]], and ''Q'' is the [[consequent]]. One statement is the '''contrapositive''' of the other only when its antecedent is the [[Negation|negated]] consequent of the other, and vice versa. The contrapositive of the example is

:<math>(\neg Q \to \neg P)</math>.

That is, "If not-''Q'', then not-''P''", or, more clearly, "If ''Q'' is not the case, then ''P'' is not the case." Using our example, this is rendered "If ''Socrates is not human'', then ''Socrates is not a man''." This statement is said to be ''contraposed'' to the original and is logically equivalent to it. Due to their logical equivalence, stating one effectively states the other; when one is [[Truth value|true]], the other is also true. Likewise with falsity.

Strictly speaking, a contraposition can only exist in two simple conditionals. However, a contraposition may also exist in two complex conditionals, if they are similar. Thus, <math>\forall{x}(P{x} \to Q{x})</math>, or "All ''P''<nowiki></nowiki>s are ''Q''<nowiki></nowiki>s," is contraposed to <math>\forall{x}(\neg Q{x} \to \neg P{x})</math>, or "All non-''Q''<nowiki></nowiki>s are non-''P''<nowiki></nowiki>s."

==Simple proof by definition of a conditional==

In [[first-order logic]], the conditional is defined as:

:<math>A \to B \iff \neg A \or B</math>

We have:

:<math>\neg A \or B \iff \neg A \or (\neg \neg B)</math>

:<math>\iff \neg (\neg B) \or \neg A </math>

:<math> \iff \neg B \to \neg A </math>

==Simple proof by contradiction==

Let:

:<math>(A \to B)\and \neg B</math>

It is given that, if A is true, then B is true, and it is also given that B is not true. We can then show that A must not be true by contradiction. For, if A were true, then B would have to also be true (given). However, it is given that B is not true, so we have a contradiction. Therefore, A is not true (assuming that we are dealing with concrete statements that are either true or not true):

:<math>(A \to B) \to (\neg B \to \neg A)</math>

We can apply the same process the other way round:

:<math>(\neg B \to \neg A)\and A</math>

We also know that B is either true or not true. If B is not true, then A is also not true. However, it is given that A is true; so, the assumption that B is not true leads to contradiction and must be false. Therefore, B must be true:

:<math>(\neg B \to \neg A) \to (A \to B)</math>

Combining the two proved statements makes them logically equivalent:

:<math>(A \to B) \iff (\neg B \to \neg A)</math>

==More rigorous proof of the equivalence of contrapositives==

Logical equivalence between two propositions means that they are true together or false together. To prove that contrapositives are [[logically equivalent]], we need to understand when material implication is true or false.

:<math>(P \to Q)</math>

This is only false when ''P'' is true and ''Q'' is false. Therefore, we can reduce this proposition to the statement "False when ''P'' and not-''Q''" (i.e. "True when it is not the case that ''P'' and not-''Q''"):

:<math>\neg(P \and \neg Q)</math>

The elements of a [[Logical conjunction|conjunction]] can be reversed with no effect (by [[commutativity]]):

:<math>\neg(\neg Q \and P)</math>

We define <math>R</math> as equal to "<math>\neg Q</math>", and <math>S</math> as equal to <math>\neg P</math> (from this, <math>\neg S</math> is equal to <math>\neg\neg P</math>, which is equal to just <math>P</math>):

:<math>\neg(R \and \neg S)</math>

This reads "It is not the case that (''R'' is true and ''S'' is false)", which is the definition of a material conditional. We can then make this substitution:

:<math>(R \to S)</math>

When we swap our definitions of ''R'' and ''S'', we arrive at the following:

:<math>(\neg Q \to \neg P)</math>

==Comparisons==
{| class="wikitable"
! name !! form !! description
|-
| implication || if ''P'' then ''Q'' || first statement implies truth of second
|-
| inverse || if not ''P'' then not ''Q'' || negation of both statements
|-
| converse || if ''Q'' then ''P'' || reversal of both statements
|-
| contrapositive || if not ''Q'' then not ''P'' || reversal and negation of both statements
|-
| negation || ''P'' and not ''Q'' || contradicts the implication
|}

===Examples===
Take the statement "''All red objects have color.''" This can be equivalently expressed as "''If an object is red, then it has color.''"

* The '''contrapositive''' is "''If an object does not have color, then it is not red.''" This follows logically from our initial statement and, like it, it is evidently true.
* The '''inverse''' is "''If an object is not red, then it does not have color.''" An object which is blue is not red, and still has color. Therefore in this case the inverse is false.
* The '''converse''' is "''If an object has color, then it is red.''" Objects can have other colors, of course, so, the converse of our statement is false.
* The '''negation''' is "''There exists a red object that does not have color.''" This statement is false because the initial statement which it negates is true.

In other words, the contrapositive is logically equivalent to a given [[Material conditional|conditional]] statement, though not sufficient for a [[biconditional]].

Similarly, take the statement "''All [[quadrilaterals]] have four sides,''" or equivalently expressed "''If a polygon is a quadrilateral, then it has four sides.''"
* The '''contrapositive''' is "''If a polygon does not have four sides, then it is not a quadrilateral.''" This follows logically, and as a rule, contrapositives share the [[truth value]] of their conditional.
* The '''inverse''' is "''If a polygon is not a quadrilateral, then it does not have four sides.''" In this case, unlike the last example, the inverse of the argument is true.
* The '''converse''' is "''If a polygon has four sides, then it is a quadrilateral.''" Again, in this case, unlike the last example, the converse of the argument is true.
* The '''negation''' is "''There is at least one quadrilateral that does not have four sides.''" This statement is clearly false.

Since the statement and the converse are both true, it is called a [[biconditional]], and can be expressed as "'''A polygon is a quadrilateral ''if, and only if,'' it has four sides.'''" (The phrase ''if and only if'' is sometimes abbreviated ''iff''.) That is, having four sides is both necessary to be a quadrilateral, and alone sufficient to deem it a quadrilateral.

===Truth===
* If a statement is true, then its contrapositive is true (and vice versa).
* If a statement is false, then its contrapositive is false (and vice versa).
* If a statement's inverse is true, then its converse is true (and vice versa).
* If a statement's inverse is false, then its converse is false (and vice versa).
* If a statement's negation is false, then the statement is true (and vice versa).
* If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, it is known as a [[logical biconditional]].

==Application==
Because the '''contrapositive''' of a statement always has the same truth value (truth or falsity) as the statement itself, it can be a powerful tool for proving mathematical [[theorem]]s via [[Reductio ad absurdum|proof by contradiction]], as in the [[Irrational_number#The_square_root_of_2|proof of the irrationality of the square root of 2]]. By the definition of a [[rational number]], the statement can be made that "''If <math>\sqrt{2}</math> is rational, then it can be expressed as an [[irreducible fraction]]''". This statement is '''true''' because it is a restatement of a true definition. The contrapositive of this statement is "''If <math>\sqrt{2}</math> cannot be expressed as an irreducible fraction, then it is not rational''". This contrapositive, like the original statement, is also '''true'''. Therefore, if it can be proven that <math>\sqrt{2}</math> cannot be expressed as an irreducible fraction, then it must be the case that <math>\sqrt{2}</math> is not a rational number.

A similar, but not identical tool for proving mathematical theorems is the [[Proof by contrapositive|proof by contraposition]].

==See also==
* ''[[Reductio ad absurdum]]''

[[Category:Mathematical logic]]
[[Category:Theorems in propositional logic]]

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