CA Foundation · Quantitative Aptitude

Sets, Relations and Functions, Limits and Continuity: formula sheet

Full chapter guide

Key formulas

Number of subsets: n(P(A)) = 2ⁿ; n is the number of elements in A. This counts the empty set and A itself.
Number of proper subsets: 2ⁿ − 1; Excludes only A itself. The empty set is still counted as a proper subset when A is non-empty.
Number of non-empty proper subsets: 2ⁿ − 2; Excludes both ∅ and A.
Subsets of fixed size: Number of subsets with r elements = nCr; Useful when a question asks for subsets with exactly 2 or 3 elements.
Equal sets: A = B if and only if A ⊆ B and B ⊆ A; Order and repetition do not matter.
Empty set rule: ∅ ⊆ A for every set A; ∅ is not the same as {0} or {∅}. {∅} has one element.
Union of two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); Subtract the overlap because it is counted twice.
Disjoint sets: If A ∩ B = ∅, then n(A ∪ B) = n(A) + n(B); A special case when there is no overlap.
Difference: n(A − B) = n(A) − n(A ∩ B); This is the 'only A' region.
Complement: n(A′) = n(U) − n(A); U is the universal set.
Neither A nor B: n(A′ ∩ B′) = n(U) − n(A ∪ B); Uses De Morgan: A′ ∩ B′ = (A ∪ B)′.
De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′; Complement is taken within the same universal set U.
Exactly one of A or B: n(A) + n(B) − 2n(A ∩ B); This is the symmetric difference: only A plus only B.
Union of three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C); Add singles, subtract pairs, add the triple.
Domain restrictions: Denominator ≠ 0; expression under an even root ≥ 0; argument of log > 0; Apply every restriction together and take the common values of x.
One-one test: f(a) = f(b) ⇒ a = b; If you can find two different inputs with the same output, it is many-one. Example: x² on all real numbers.
Onto test: Range = Codomain; Always check what the codomain is. The same rule can be onto for one codomain and into for another.
Composite function: (f∘g)(x) = f(g(x)); Work inside out. In general f∘g ≠ g∘f.
Inverse function: f⁻¹(f(x)) = x and f(f⁻¹(y)) = y; Exists only if f is one-one and onto. f⁻¹(x) is not 1 ÷ f(x).
Inverse of a linear function: If f(x) = ax + b (a ≠ 0), then f⁻¹(x) = (x − b) ÷ a; Swap x and y, then solve for y.
Even function: f(−x) = f(x); Examples: x², |x|, constant.
Odd function: f(−x) = −f(x); Examples: x, x³. If 0 is in the domain, an odd function has f(0) = 0.
Counting functions: If A has m elements and B has n elements: total functions = nᵐ; one-one functions = n(n−1)…(n−m+1), needs n ≥ m; bijections (m = n) = n!; Useful for quick MCQs on the number of functions.
Even and odd combinations: even ± even = even; odd ± odd = odd; even × even = even; odd × odd = even; even × odd = odd; Use these to classify a function without calculation.
Existence of a limit: lim(x→a) f(x) exists ⇔ LHL = RHL (both finite); LHL is the limit as x → a⁻. RHL is the limit as x → a⁺.
Sum and difference: lim [f(x) ± g(x)] = lim f(x) ± lim g(x); Valid when both individual limits exist.
Product and constant multiple: lim [f(x)·g(x)] = lim f(x) · lim g(x); lim [k·f(x)] = k · lim f(x); Valid when the individual limits exist.
Quotient: lim [f(x) ÷ g(x)] = lim f(x) ÷ lim g(x); Valid only when lim g(x) ≠ 0.
Power-difference limit: lim(x→a) (xⁿ − aⁿ) ÷ (x − a) = n·aⁿ⁻¹; Holds for any real n when a > 0. For a positive integer n it holds for all real a.
Exponential limit: lim(x→0) (eˣ − 1) ÷ x = 1; More generally, lim(x→0) (eᵃˣ − 1) ÷ x = a.
Logarithmic limit: lim(x→0) log(1 + x) ÷ x = 1; Here log means natural log (base e). More generally, lim(x→0) log(1 + ax) ÷ x = a.
General exponential limit: lim(x→0) (aˣ − 1) ÷ x = log a; Valid for a > 0. Log is natural log.
Definition of e: lim(x→0) (1 + x)^(1/x) = e; lim(x→∞) (1 + 1/x)ˣ = e; Useful for forms like (1 + kx)^(1/x), which tends to eᵏ as x → 0.
Limit at infinity of a ratio of polynomials: Compare the highest powers of x in numerator and denominator; Equal degree gives the ratio of leading coefficients. Higher degree in the denominator gives 0.
Condition for continuity at x = a: lim (x→a) f(x) = f(a); This needs f(a) defined and the limit to exist, so LHL = RHL = f(a).
One-sided limits: LHL = lim (x→a⁻) f(x), RHL = lim (x→a⁺) f(x); The limit exists only when LHL = RHL (both finite).
Removable discontinuity: lim (x→a) f(x) exists, but f(a) is undefined or ≠ the limit; Can be removed by defining f(a) equal to the limit.
Jump discontinuity: LHL ≠ RHL (both finite); The limit does not exist. It cannot be removed.
Finding an unknown constant: LHL = RHL = f(a); Put the boundary value into each piece, equate, and solve.
Standard factor cancellation: (x² − a²) ÷ (x − a) = x + a, for x ≠ a; The most common way to get a limit of a 0/0 form.

Quick revision

A set is a well-defined collection; the empty set has no elements and is a subset of every set.
A set with n elements has 2ⁿ subsets.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
A relation from A to B is a subset of A × B; if A has m elements and B has n, then n(A × B) = mn.
Reflexive: (a, a) is in R for every a. Symmetric: (a, b) in R implies (b, a) in R. Transitive: (a, b) and (b, c) in R imply (a, c) in R.
An equivalence relation is reflexive, symmetric and transitive.
A function gives exactly one output for each input; no input can be left out or have two outputs.
One-one means different inputs give different outputs; onto means the range equals the codomain.
If the limits of f and g both exist (finite), then the limit of the sum, product or quotient equals the sum, product or quotient of the limits (quotient needs a non-zero denominator limit).
For 0/0 forms, factorise and cancel before substituting again.
lim (x→0) (eˣ − 1)/x = 1 and lim (x→0) log(1 + x)/x = 1.
f is continuous at x = a if left limit = right limit = f(a).

Common mistakes

Counting repeated elements while finding n. Fix: Cross out repeats first. {1, 2, 2, 3} has n = 3, so 2ⁿ = 8.
Using 2ⁿ when the question asks for proper subsets. Fix: Underline the word proper. Proper subsets = 2ⁿ − 1, because A itself is excluded.
Forgetting to subtract the overlap in n(A ∪ B). Fix: Always write the full formula first. Ask whether anyone is in both.
Treating 'only A' as n(A). Fix: Use n(A) − n(A ∩ B) for only A.
Reversing the order in a composite function, computing f(g(x)) as g(f(x)). Fix: Always say 'inner first'. In f(g(x)), g sits inside the bracket, so it is done first.
Writing f⁻¹(x) = 1 ÷ f(x). Fix: f⁻¹ means the inverse function. Swap x and y and solve. Check with f(f⁻¹(x)) = x.
Writing the answer as 0 or 'undefined' when substitution gives 0/0. Fix: Treat 0/0 as a prompt. Factorise, rationalise or use a standard limit before substituting again.
Cancelling terms instead of factors, such as cancelling x in (x + 2)/(x + 3). Fix: Factorise fully. Cancel only a factor common to the whole numerator and the whole denominator.
Checking only that the limit exists and ignoring f(a). Fix: Always write all three values: LHL, RHL and f(a). Continuity needs all three equal.
Cancelling (x − a) and then saying f(a) equals the cancelled expression. Fix: Use the cancelled form only to find the limit. f(a) comes from the given definition at a.

Exam tips

Read the exact wording: all subsets, proper subsets, and non-empty proper subsets give three different answers, and wrong options are built around them.
In set-builder questions, list the elements first. Most mistakes come from guessing the count without listing.
Learn the powers of 2 up to 2¹⁰ = 1,024 so you do not waste time on multiplication.
For true/false statements, test with {1, 2} or with ∅ rather than arguing in general.
With 0.25 negative marking, guess only after you have removed at least one option using a quick check.
Read the wording carefully. 'Either A or B' may mean at least one, and 'only' means the overlap must be removed.
If the numbers cannot be placed without a negative region, the data is inconsistent. Recheck your reading rather than force an answer.
For set-listing questions, write out the actual elements. Do not trust mental work. Paper 3 (Quantitative Aptitude) has negative marking of 0.25 per wrong answer, so careless errors cost marks.