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The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
In mathematics, theorems are often stated in the form "P is true if and only if Q is true". Because, as explained in previous section, necessity of one for the other is equivalent to sufficiency of the other for the first one, e.g. P ⇐ Q {\displaystyle P\Leftarrow Q} is equivalent to Q ⇒ P {\displaystyle Q\Rightarrow P} , if P is necessary ...
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally ...
The law of identity can be expressed as (=), where x is a variable ranging over the domain of all individuals. In logic, there are various different ways identity can be handled. In first-order logic with identity, identity is treated as a logical constant and its axioms are part of the logic itself. Under this convention, the law of identity ...
The material conditional (also known as material implication) is an operation commonly used in logic.When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false.
A formula is logically valid (or simply valid) if it is true in every interpretation. [22] These formulas play a role similar to tautologies in propositional logic. A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.