The inductive effect is the permanent displacement of the shared sigma-bond electron pair towards the more electronegative atom, transmitted through a chain of carbon atoms and decreasing with distance. Groups that push electrons away from themselves (electron-releasing, e.g. alkyl groups -CH3) show the +I effect, while groups that pull electrons towards themselves (electron-withdrawing, e.g. -NO2, -Cl) show the -I effect.

Resonance is the representation of a molecule or ion by two or more structures (canonical/contributing structures) that differ only in the position of electrons, none of which alone describes it fully; the actual structure is a resonance hybrid. Resonance energy is the difference in energy between the resonance hybrid and the most stable contributing structure; the greater the resonance energy, the more stable the molecule. Example: benzene, which is a hybrid of two Kekule structures and is more stable than expected.

An electrophile (electron-loving) is an electron-deficient species that accepts a pair of electrons; examples: H+, NO2+ (also positively charged or neutral electron-deficient species like BF3). A nucleophile (nucleus-loving) is an electron-rich species that donates a pair of electrons; examples: OH-, CN- (also neutral species with lone pairs like NH3 and H2O).

In homolytic fission, a covalent bond breaks so that each bonded atom takes one of the shared electrons, producing neutral free radicals (species with an unpaired electron); it usually occurs in the presence of heat or light in non-polar bonds. In heterolytic fission, the bond breaks so that one atom takes both shared electrons, producing oppositely charged ions (a cation and an anion, e.g. a carbocation and an anion).

Wurtz reaction: two molecules of an alkyl halide react with sodium metal in dry ether to give a symmetrical alkane with double the number of carbon atoms, e.g. 2CH3Cl + 2Na -> CH3-CH3 + 2NaCl (ethane). Decarboxylation: the sodium salt of a carboxylic acid is heated with soda lime (NaOH + CaO) to give an alkane with one carbon less, e.g. CH3COONa + NaOH -> CH4 + Na2CO3 (methane).

2CH3Cl + 2Na -> C2H6 + 2NaCl

Markovnikov's rule states that when an unsymmetrical reagent (HX) adds to an unsymmetrical alkene, the negative part of the reagent (X) attaches to the carbon bearing fewer hydrogen atoms, while hydrogen adds to the carbon bearing more hydrogen atoms. So HBr adds to propene (CH3-CH=CH2) to give mainly 2-bromopropane (CH3-CHBr-CH3), because the more stable secondary carbocation is formed. In the presence of peroxide, addition is anti-Markovnikov (peroxide/Kharasch effect), giving 1-bromopropane.

CH3-CH=CH2 + HBr -> CH3-CHBr-CH3

Ozonolysis is the reaction of an alkene with ozone (O3) to form an unstable ozonide, which on reductive cleavage (with Zn and water) breaks the carbon-carbon double bond and gives carbonyl compounds (aldehydes and/or ketones). It is used to locate the position of the double bond. Propene (CH3-CH=CH2) on ozonolysis gives ethanal (CH3CHO) and methanal (HCHO).

CH3-CH=CH2 -> CH3CHO + HCHO

Benzene has a stable, delocalised aromatic pi-electron cloud (6 pi electrons). Addition reactions would destroy this aromatic stability, so benzene prefers substitution, which preserves the aromatic ring. The pi cloud attracts electrophiles, so it undergoes electrophilic substitution. Examples: nitration (with conc. HNO3 + conc. H2SO4, electrophile NO2+) and halogenation (with Cl2/FeCl3); others are sulphonation and Friedel-Crafts alkylation/acylation.

The power set is the set of all subsets of A. P(A) = { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }. The number of elements n(P(A)) = 2^n = 2^3 = 8.

n(P(A)) = 2^n

Let n(C) = 24, n(F) = 16, n(C union F) = 35 (each likes at least one). By the formula n(C union F) = n(C) + n(F) - n(C intersection F), we get 35 = 24 + 16 - n(C intersection F), so n(C intersection F) = 40 - 35 = 5. Hence 5 students like both games.

n(A union B) = n(A) + n(B) - n(A intersection B)

Union (A union B) is the set of all elements that are in A or in B or in both. Intersection (A intersection B) is the set of elements common to both A and B. Difference (A - B) is the set of elements that are in A but not in B. Example: if A = {1,2,3} and B = {2,3,4}, then A union B = {1,2,3,4}, A intersection B = {2,3}, and A - B = {1}.

{x : -3 <= x < 5} includes -3 but not 5, so in interval notation it is [-3, 5). The interval (2, 9] excludes 2 and includes 9, so in set-builder form it is {x : x is a real number, 2 < x <= 9}.

The Cartesian product A x B is the set of all ordered pairs (a, b) with a in A and b in B. A x B = {(1,3), (1,4), (2,3), (2,4)}. The number of elements n(A x B) = n(A) x n(B) = 2 x 2 = 4.

n(A x B) = n(A) x n(B)

Two ordered pairs are equal when their corresponding components are equal. So x + 1 = 3 gives x = 2, and y - 2 = 1 gives y = 3. Hence x = 2 and y = 3.

(a, b) = (c, d) => a = c and b = d

The domain of a relation is the set of all first components (inputs), and the range is the set of all second components (outputs). For R = {(1,2), (2,4), (3,6)}, the domain = {1, 2, 3} and the range = {2, 4, 6}.

A function f from A to B is one-one (injective) if different elements of A have different images in B, i.e. f(x1) = f(x2) implies x1 = x2. A function f from A to B is onto (surjective) if every element of B is the image of at least one element of A, i.e. the range of f equals the codomain B.

Using pi radians = 180 degrees: 5pi/6 radians = (5/6) x 180 = 150 degrees. And 240 degrees = 240 x (pi/180) = 4pi/3 radians.

radians = degrees x (pi/180)

Write 75 = 45 + 30. Using sin(A + B) = sin A cos B + cos A sin B: sin 75 = sin 45 cos 30 + cos 45 sin 30 = (1/sqrt2)(sqrt3/2) + (1/sqrt2)(1/2) = (sqrt3 + 1)/(2 sqrt2) = (sqrt6 + sqrt2)/4.

sin(A + B) = sin A cos B + cos A sin B

Start from the fundamental identity sin^2 x + cos^2 x = 1. Divide both sides by cos^2 x (cos x not equal to 0): sin^2 x/cos^2 x + cos^2 x/cos^2 x = 1/cos^2 x, which gives tan^2 x + 1 = sec^2 x. Hence 1 + tan^2 x = sec^2 x.

1 + tan^2 x = sec^2 x

Write 15 = 45 - 30. Using sin(A - B) = sin A cos B - cos A sin B: sin 15 = sin 45 cos 30 - cos 45 sin 30 = (1/sqrt2)(sqrt3/2) - (1/sqrt2)(1/2) = (sqrt3 - 1)/(2 sqrt2) = (sqrt6 - sqrt2)/4.

sin(A - B) = sin A cos B - cos A sin B