DSST Math for Liberal Arts cheat sheet

Hierarchy: naturals (1, 2, 3, …) ⊂ whole numbers (adds 0) ⊂ integers (adds negatives) ⊂ rationals (a/b with integers a, b and b ≠ 0) ⊂ reals (adds irrationals such as √2 and π).

A number is rational exactly when its decimal terminates or repeats; non-terminating, non-repeating decimals are irrational.

Absolute value |x| is distance from 0 on the number line, so |x| ≥ 0 always, and |a − b| is the distance between a and b.

Order of operations: parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right.

Sign rules: negative × negative = positive; negative × positive = negative; subtracting a negative is adding, a − (−b) = a + b.

Rules of signs for fractions and division follow multiplication: (−a)/(−b) = a/b and (−a)/b = −(a/b).

To compare fractions, cross-multiply: a/b < c/d exactly when ad < bc (for positive denominators).

Percent means per hundred: 15% = 0.15 = 3/20. To convert a fraction to a percent, divide, then multiply by 100.

For any real a, √(a²) = |a|, not a; squaring and square-rooting are inverse only for non-negative inputs.

Set operations: A ∪ B contains elements in A or B (or both); A ∩ B contains elements in both; A′ (complement) contains elements of the universal set not in A.

Inclusion–exclusion: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets subtract each pairwise intersection and add back the triple intersection.

De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

A ⊆ B means every element of A is in B; the empty set ∅ is a subset of every set, and a set with n elements has 2ⁿ subsets.

Conditional p → q is false only when p is true and q is false; it is equivalent to its contrapositive ∼q → ∼p.

Converse of p → q is q → p and is not equivalent to the original; inverse is ∼p → ∼q, also not equivalent.

Negation rules: ∼(p ∧ q) ≡ ∼p ∨ ∼q and ∼(p ∨ q) ≡ ∼p ∧ ∼q.

p → q is equivalent to ∼p ∨ q; p ↔ q is true exactly when p and q have the same truth value.

Venn-diagram counting: fill the innermost region first, then work outward using totals given for each region.

Metric prefixes: kilo = 1000, hecto = 100, deka = 10, base unit, deci = 0.1, centi = 0.01, milli = 0.001.

Moving one prefix step left multiplies by 10; one step right divides by 10. Example: 3.5 km = 3500 m = 350,000 cm.

Mass vs. volume: 1 cm³ = 1 mL and 1000 cm³ = 1 L; 1 kg of water occupies 1 L at standard density.

Temperature: °F = (9/5)°C + 32 and °C = (5/9)(°F − 32); K = °C + 273.15.

Perimeter and area: rectangle P = 2l + 2w, A = lw; triangle A = ½bh; circle C = 2πr, A = πr².

Trapezoid A = ½(b₁ + b₂)h; parallelogram A = bh; volume of a rectangular prism V = lwh, cylinder V = πr²h.

Pythagorean theorem a² + b² = c² for right triangles; the hypotenuse c is opposite the right angle.

Similar figures: corresponding angles are equal and side lengths share one scale factor k; areas scale by k² and volumes by k³.

Angle facts: straight angle = 180°, triangle angles sum to 180°, and a quadrilateral's interior angles sum to 360°.

Slope m = (y₂ − y₁)/(x₂ − x₁); positive slope rises, negative falls, zero is horizontal, undefined is vertical.

Line forms: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁); standard Ax + By = C.

Parallel lines have equal slopes; perpendicular slopes are negative reciprocals (m₁m₂ = −1).

A function assigns exactly one output per input; the vertical line test fails a graph that is not a function.

For f(x) = ax² + bx + c, the vertex is at x = −b/(2a); a > 0 opens upward, a < 0 opens downward.

Function composition (f ∘ g)(x) = f(g(x)); evaluate inside-out, and the domain of g must be respected.

Direct variation y = kx gives a line through the origin; inverse variation y = k/x gives a hyperbola.

Average rate of change of f on [a, b] is (f(b) − f(a))/(b − a), the slope of the secant line.

To find an x-intercept set y = 0; to find a y-intercept set x = 0.

A system of two linear equations has one solution (different slopes), none (same slope, different intercepts), or infinitely many (identical lines).

Substitution: solve one equation for a variable, substitute into the other, then back-substitute.

Elimination: multiply equations so one variable's coefficients are opposites, add, solve, then back-substitute.

Graphically, the solution of a system is the intersection point of the lines; check by substituting both values.

Solving a linear inequality: add or subtract freely; multiplying or dividing by a negative reverses the inequality sign.

The solution of a two-variable inequality is a half-plane; a solid boundary line includes the line, a dashed one excludes it.

Test a point (often the origin) to decide which side of the boundary to shade.

Compound inequalities: |x| < a means −a < x < a; |x| > a means x < −a or x > a.

A system of inequalities is solved by graphing and shading the overlap of all half-planes.

Exponent rules: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

Fractional exponents: a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(aᵐ); also (ab)ⁿ = aⁿbⁿ.

Log definition: log_b x = y means bʸ = x, with b > 0, b ≠ 1, x > 0; ln x means log_e x.

Log laws: log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n log x, log_b b = 1, log_b 1 = 0.

Change of base: log_b x = (log x)/(log b), allowing evaluation with any calculator base.

Compound interest A = P(1 + r/n)^(nt); continuous compounding A = Pe^(rt).

Simple interest I = Prt, and total amount A = P(1 + rt).

Doubling time at rate r with continuous compounding is t = ln 2 / r; exponential growth/decay is y = a·bᵗ.

Rule of 72: doubling time in years is about 72 divided by the annual percentage rate.

Fundamental counting principle: if choices are made in stages, multiply the number of options at each stage.

Permutations order matters: nPr = n!/(n − r)!; combinations order does not matter: nCr = n!/(r!(n − r)!).

n! counts arrangements of n distinct objects; 0! = 1 and 1! = 1.

Permutations of a multiset with repeats: n!/(n₁!·n₂!·…), e.g. arrangements of the letters in BANANA.

Circular arrangements of n distinct objects number (n − 1)!; seating around a round table is the standard case.

Addition rule: if two cases cannot both happen, add their counts; if they can overlap, subtract the overlap.

Arrangements with repetition allowed: nʳ ways to fill r positions from n choices, e.g. 3-digit codes from 10 digits = 10³.

Combinations with repetition (stars and bars) count nCr(n + r − 1, r) ways to distribute r identical items into n bins.

At least one: count the complement and subtract from the total, e.g. hands with at least one ace = total − hands with none.

Probability P(E) = favorable outcomes / total outcomes, with 0 ≤ P(E) ≤ 1; P(not E) = 1 − P(E).

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events subtract nothing.

Multiplication rule: P(A and B) = P(A)·P(B|A); independent events satisfy P(A and B) = P(A)·P(B).

Conditional probability P(B|A) = P(A and B)/P(A).

Expected value = Σ (value × probability); it need not be an attainable outcome.

Mean = sum/n; median = middle value of ordered data; mode = most frequent value.

Standard deviation measures spread about the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score z = (x − μ)/σ states how many standard deviations a value lies from the mean.

Counting in probability: when outcomes are equally likely, use permutations and combinations to size the numerator and denominator.

Hierarchy: naturals ⊂ whole numbers ⊂ integers ⊂ rationals ⊂ reals; irrationals such as √2 and π sit in the reals but not the rationals.

Rational numbers have terminating or repeating decimals; irrational decimals never terminate or repeat.

Absolute value |x| is distance from 0, so |a − b| is the distance between a and b on the number line.

Order of operations: parentheses, exponents, multiplication/division left to right, addition/subtraction left to right.

Sign rules: negative × negative = positive; a − (−b) = a + b; (−a)/(−b) = a/b.

Compare fractions by cross-multiplying: a/b < c/d exactly when ad < bc for positive denominators.

Percent means per hundred: 15% = 0.15 = 3/20; to convert a fraction to a percent, divide then multiply by 100.

√(a²) = |a| for every real a, so squaring then square-rooting does not simply return a.

A ∪ B is the union (in either set); A ∩ B is the intersection (in both); A′ is the complement within the universal set.

Inclusion–exclusion: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets subtract pairwise intersections and add the triple intersection back.

De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

A set with n elements has 2ⁿ subsets; ∅ is a subset of every set.

p → q is false only when p is true and q is false; it is equivalent to ∼p ∨ q and to its contrapositive ∼q → ∼p.

Converse (q → p) and inverse (∼p → ∼q) are not equivalent to the original conditional.

Negation rules: ∼(p ∧ q) ≡ ∼p ∨ ∼q and ∼(p ∨ q) ≡ ∼p ∧ ∼q.

p ↔ q is true exactly when p and q share the same truth value.

Venn counting: fill the innermost overlap first, then work outward from the given region totals.

Prefixes: kilo 1000, hecto 100, deka 10, deci 0.1, centi 0.01, milli 0.001.

Each prefix step left multiplies by 10; each step right divides by 10, so 3.5 km = 3500 m = 350,000 cm.

1 cm³ = 1 mL and 1000 cm³ = 1 L; 1 kg of water occupies about 1 L.

Temperature: °F = (9/5)°C + 32; °C = (5/9)(°F − 32); K = °C + 273.15.

Rectangle P = 2l + 2w and A = lw; triangle A = ½bh; circle C = 2πr and A = πr².

Trapezoid A = ½(b₁ + b₂)h; rectangular prism V = lwh; cylinder V = πr²h.

Pythagorean theorem a² + b² = c² applies only to right triangles, with c the hypotenuse.

Similar figures scale lengths by k, areas by k², and volumes by k³.

Triangle angles sum to 180°, a straight angle is 180°, and quadrilateral interior angles sum to 360°.

Slope m = (y₂ − y₁)/(x₂ − x₁); zero slope is horizontal, undefined slope is vertical.

Line forms: y = mx + b (slope-intercept), y − y₁ = m(x − x₁) (point-slope), Ax + By = C (standard).

Parallel lines have equal slopes; perpendicular slopes multiply to −1.

A function gives exactly one output per input; the vertical line test detects non-functions.

For f(x) = ax² + bx + c the vertex is at x = −b/(2a); a > 0 opens up, a < 0 opens down.

Composition (f ∘ g)(x) = f(g(x)) is evaluated inside-out, with x restricted to g's domain.

Direct variation y = kx is a line through the origin; inverse variation y = k/x is a hyperbola.

Average rate of change on [a, b] is (f(b) − f(a))/(b − a), the secant-line slope.

Intercepts: set y = 0 for x-intercepts, set x = 0 for the y-intercept.

Two linear equations give one solution (different slopes), no solution (same slope, different intercepts), or infinitely many (same line).

Substitution: isolate one variable, substitute into the other equation, solve, then back-substitute.

Elimination: scale equations so one variable's coefficients are opposites, add, solve, then back-substitute.

Graphically the solution of a system is the lines' intersection point; verify by substituting both values.

Multiplying or dividing an inequality by a negative number reverses its direction.

A two-variable inequality shades a half-plane; a solid boundary includes the line, a dashed boundary excludes it.

Test a point such as the origin to choose which side of the boundary to shade.

Absolute value: |x| < a means −a < x < a; |x| > a means x < −a or x > a.

A system of inequalities is solved by shading each half-plane and taking the overlap.

Rules: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

Fractional exponents: a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(aᵐ); also (ab)ⁿ = aⁿbⁿ.

log_b x = y means bʸ = x, requiring b > 0, b ≠ 1, x > 0; ln x is log base e.

Log laws: log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n log x, log_b 1 = 0.

Change of base: log_b x = (log x)/(log b).

Compound interest A = P(1 + r/n)^(nt); continuous compounding A = Pe^(rt).

Simple interest I = Prt with total A = P(1 + rt).

Exponential growth or decay y = a·bᵗ; continuous doubling time t = ln 2 / r, and the Rule of 72 estimates it as 72 divided by the percent rate.

Fundamental counting principle: multiply the number of options at each successive stage.

nPr = n!/(n − r)! counts ordered selections; nCr = n!/(r!(n − r)!) counts unordered selections.

n! counts arrangements of n distinct objects; 0! = 1.

Arrangements of a multiset with repeats: n!/(n₁!·n₂!·…), as with the letters of BANANA.

Circular arrangements of n distinct objects number (n − 1)!.

Addition rule: add counts of non-overlapping cases; subtract the overlap when cases can occur together.

With repetition allowed, r positions from n choices give nʳ arrangements, e.g. 10³ three-digit codes.

Stars and bars: nCr(n + r − 1, r) ways to distribute r identical items among n bins.

For 'at least one', count the complement and subtract from the total.

P(E) = favorable/total with 0 ≤ P(E) ≤ 1, and P(not E) = 1 − P(E).

P(A or B) = P(A) + P(B) − P(A and B); mutually exclusive events have P(A and B) = 0.

P(A and B) = P(A)·P(B|A); independent events give P(A and B) = P(A)·P(B).

Conditional probability P(B|A) = P(A and B)/P(A).

Expected value = Σ(value × probability) and need not equal any possible outcome.

Mean = sum/n; median = middle of ordered data; mode = most frequent value.

Standard deviation measures spread; roughly 68% of normal data lies within 1 SD and 95% within 2 SD of the mean.

z-score z = (x − μ)/σ gives the number of standard deviations from the mean.

Equally likely outcomes: use permutations and combinations to count the numerator and denominator.

Naturals are counting numbers; whole numbers add 0; integers add negatives; rationals are ratios of integers; reals include irrationals.

A rational number's decimal terminates or repeats; an irrational decimal does neither.

|x| is the distance from 0, so |a − b| gives the distance between a and b.

Order of operations: parentheses, exponents, multiplication/division left to right, then addition/subtraction.

Subtracting a negative adds: a − (−b) = a + b; the product of two negatives is positive.

Cross-multiply to compare fractions: a/b < c/d exactly when ad < bc for positive denominators.

Percent means per hundred, so 15% = 0.15 = 3/20; fraction to percent: divide, then multiply by 100.

√(a²) = |a| for all real a, not a itself.

Union A ∪ B: in A or B; intersection A ∩ B: in both; complement A′: in the universal set but not A.

Inclusion–exclusion n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets subtract pairwise intersections and add the triple back.

De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

A set with n elements has 2ⁿ subsets, and ∅ is a subset of every set.

p → q is false only when p is true and q is false; it equals ∼p ∨ q and its contrapositive ∼q → ∼p.

Converse q → p and inverse ∼p → ∼q are not equivalent to the original conditional.

∼(p ∧ q) ≡ ∼p ∨ ∼q and ∼(p ∨ q) ≡ ∼p ∧ ∼q.

p ↔ q is true exactly when p and q have the same truth value.

Venn counting: fill the central overlap first, then work outward using the stated region totals.

Prefixes: kilo 1000, hecto 100, deka 10, deci 0.1, centi 0.01, milli 0.001.

Each prefix step left multiplies by 10 and each step right divides by 10, so 3.5 km = 3500 m = 350,000 cm.

1 cm³ = 1 mL, 1000 cm³ = 1 L, and 1 kg of water occupies about 1 L.

Temperature: °F = (9/5)°C + 32; °C = (5/9)(°F − 32); K = °C + 273.15.

Rectangle P = 2l + 2w, A = lw; triangle A = ½bh; circle C = 2πr, A = πr².

Trapezoid A = ½(b₁ + b₂)h; prism V = lwh; cylinder V = πr²h.

Pythagorean theorem a² + b² = c² holds only for right triangles, with c the hypotenuse.

Similar figures scale length by k, area by k², and volume by k³.

Triangle angles sum to 180°, a straight angle is 180°, and quadrilateral interior angles sum to 360°.

Slope m = (y₂ − y₁)/(x₂ − x₁); horizontal lines have slope 0 and vertical lines have undefined slope.

Line forms: y = mx + b, y − y₁ = m(x − x₁), and Ax + By = C.

Parallel lines share a slope; perpendicular slopes multiply to −1.

A function has exactly one output per input, detected by the vertical line test.

For f(x) = ax² + bx + c the vertex is at x = −b/(2a); a > 0 opens up and a < 0 opens down.

Composition (f ∘ g)(x) = f(g(x)) is evaluated inside-out within g's domain.

Direct variation y = kx is a line through the origin; inverse variation y = k/x is a hyperbola.

Average rate of change on [a, b] is (f(b) − f(a))/(b − a), the secant slope.

Set y = 0 for x-intercepts and x = 0 for the y-intercept.

Two lines give one solution (different slopes), none (same slope, different intercepts), or infinitely many (same line).

Substitution: isolate a variable, substitute into the other equation, solve, then back-substitute.

Elimination: scale equations to opposite coefficients, add, solve, then back-substitute.

The graphical solution of a system is the intersection point; verify by substituting both values.

Multiplying or dividing an inequality by a negative reverses the sign.

A two-variable inequality shades a half-plane; solid boundary includes the line, dashed excludes it.

Test a point such as the origin to pick the side to shade.

|x| < a means −a < x < a; |x| > a means x < −a or x > a.

A system of inequalities is solved by shading each half-plane and taking the common overlap.

aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ⁿ = 1/aⁿ.

a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(aᵐ); also (ab)ⁿ = aⁿbⁿ.

log_b x = y means bʸ = x with b > 0, b ≠ 1, x > 0; ln x is log base e.

log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n log x, log_b 1 = 0.

Change of base: log_b x = (log x)/(log b).

A = P(1 + r/n)^(nt) for compound interest; A = Pe^(rt) for continuous compounding.

Simple interest I = Prt, so the total is A = P(1 + rt).

Growth/decay y = a·bᵗ; continuous doubling time t = ln 2 / r, and the Rule of 72 estimates it as 72 divided by the percent rate.

Fundamental counting principle: multiply the options available at each successive stage.

nPr = n!/(n − r)! counts ordered selections; nCr = n!/(r!(n − r)!) counts unordered ones.

n! counts arrangements of n distinct objects, and 0! = 1.

Multiset arrangements: n!/(n₁!·n₂!·…), as with the letters of BANANA.

Circular arrangements of n distinct objects number (n − 1)!.

Add counts of non-overlapping cases; subtract the overlap when cases can occur together.

With repetition allowed, r positions from n choices give nʳ arrangements, e.g. 10³ three-digit codes.

Stars and bars: nCr(n + r − 1, r) ways to place r identical items into n bins.

For 'at least one', count the complement and subtract it from the total.

P(E) = favorable/total with 0 ≤ P(E) ≤ 1, and P(not E) = 1 − P(E).

P(A or B) = P(A) + P(B) − P(A and B); mutually exclusive events have P(A and B) = 0.

P(A and B) = P(A)·P(B|A); independent events satisfy P(A and B) = P(A)·P(B).

Conditional probability P(B|A) = P(A and B)/P(A).

Expected value = Σ(value × probability) and may not equal any attainable outcome.

Mean = sum/n; median = middle of ordered data; mode = most frequent value.

Roughly 68% of normal data lies within 1 SD of the mean and 95% within 2 SD.

z-score z = (x − μ)/σ counts standard deviations from the mean.

For equally likely outcomes, use permutations and combinations to size the numerator and denominator.