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Universal generalization / instantiation. Existential generalization / instantiation. In propositional logic and Boolean algebra, De Morgan's laws, [ 1][ 2][ 3] also known as De Morgan's theorem, [ 4] are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British ...
Elementary algebra. Propositional calculus. In mathematics, the associative property[ 1] is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs .
Mathematical fallacy. In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known ...
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more ...
The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables. The proof holds in any commutative ring. Conversely, if this identity holds in a ring R for all pairs of elements a and b, then R is commutative. To see this, apply the distributive law to ...
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 ...
Rational root theorem. In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation with integer coefficients and . Solutions of the equation are also called roots or zeros of the polynomial on the left side.
Idempotence ( UK: / ˌɪdɛmˈpoʊtəns /, [ 1] US: / ˈaɪdəm -/) [ 2] is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in ...
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