Proof by contradictionIn 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 valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
HydrideIn chemistry, a hydride is formally the anion of hydrogen (H−), a hydrogen atom with two electrons. The term is applied loosely. At one extreme, all compounds containing covalently bound H atoms are called hydrides: water (H2O) is a hydride of oxygen, ammonia is a hydride of nitrogen, etc. For inorganic chemists, hydrides refer to compounds and ions in which hydrogen is covalently attached to a less electronegative element. In such cases, the H centre has nucleophilic character, which contrasts with the protic character of acids.
Functional groupIn organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule's characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the rest of the molecule's composition. This enables systematic prediction of chemical reactions and behavior of chemical compounds and the design of chemical synthesis. The reactivity of a functional group can be modified by other functional groups nearby.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Group 14 hydrideGroup 14 hydrides are chemical compounds composed of hydrogen atoms and group 14 atoms (the elements of group 14 are carbon, silicon, germanium, tin, lead and flerovium). The tetrahydride series has the chemical formula , with X representing any of the carbon family. Methane is commonly the result of the decomposition of organic matter and is a greenhouse gas. The other hydrides are generally unstable, poisonous metal hydrides. They take on a pyramidal structure, and as such are not polar molecules like the other p-block hydrides.
Aluminium hydrideAluminium hydride (also known as alane and alumane) is an inorganic compound with the formula AlH3. Alane and its derivatives are common reducing (hydride addition) reagents in organic synthesis that are used in solution at both laboratory and industrial scales. In solution—typically in etherial solvents such tetrahydrofuran or diethyl ether—aluminium hydride forms complexes with Lewis bases, and reacts selectively with particular organic functional groups (e.g.
Proof by exhaustionProof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: A proof that the set of cases is exhaustive; i.e.
Organic redox reactionOrganic reductions or organic oxidations or organic redox reactions are redox reactions that take place with organic compounds. In organic chemistry oxidations and reductions are different from ordinary redox reactions, because many reactions carry the name but do not actually involve electron transfer. Instead the relevant criterion for organic oxidation is gain of oxygen and/or loss of hydrogen, respectively. Simple functional groups can be arranged in order of increasing oxidation state.
Mathematical inductionMathematical induction is a method for proving that a statement is true for every natural number , that is, that the infinitely many cases all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). A proof by induction consists of two cases.
Lithium aluminium hydrideLithium aluminium hydride, commonly abbreviated to LAH, is an inorganic compound with the chemical formula or . It is a white solid, discovered by Finholt, Bond and Schlesinger in 1947. This compound is used as a reducing agent in organic synthesis, especially for the reduction of esters, carboxylic acids, and amides. The solid is dangerously reactive toward water, releasing gaseous hydrogen (H2). Some related derivatives have been discussed for hydrogen storage. LAH is a colourless solid but commercial samples are usually gray due to contamination.
