Relations and Functions — JEE Mains Mathematics

Formulas (3)

• Sets given by conditions, then combined · Difference of sets
  $$A-B=A\cap B'$$
• Counting overlapping groups by inclusion–exclusion · Three sets
  $$n(A\cup B\cup C)=\textstyle\sum n(A)-\sum n(A\cap B)+n(A\cap B\cap C)$$
• Counting subsets · Subsets meeting a k-element part
  $$2^n-2^{n-k}$$

Watch out for (3)

• Strict or not at the ends
  → Sets given by conditions, then combined
• The least overlap is not zero
  → Counting overlapping groups by inclusion–exclusion
• Only the shared elements count
  → Counting subsets

Formulas (2)

• Testing reflexive, symmetric and transitive · Transitivity
  $$aRb\ \text{and}\ bRc\ \Rightarrow\ aRc$$
• Equivalence relations and their classes · The usual shape
  $$aRb\iff g(a)=g(b)$$

Watch out for (2)

• A chain back to the start
  → Testing reflexive, symmetric and transitive
• Reflexive and symmetric is not enough
  → Equivalence relations and their classes

Formulas (3)

• Counting the pairs in a relation · Pairs in a relation
  $$n(R)=\sum_{a}\#\{b: aRb\}$$
• Fewest pairs to add: reflexive, symmetric, equivalence · Smallest equivalence containing R
  $$\sum_i k_i^2\ \text{pairs, } k_i=\text{class sizes}$$
• Counting relations of a given type · Symmetric relations
  $$2^{n(n+1)/2}$$

Watch out for (3)

• Count both orders
  → Counting the pairs in a relation
• Symmetry then transitivity brings the diagonal
  → Fewest pairs to add: reflexive, symmetric, equivalence
• Equivalences are not a power of 2
  → Counting relations of a given type

Formulas (3)

• The conditions that fix a domain · Inverse-sine domain
  $$\sin^{-1}u,\ \cos^{-1}u:\ -1\le u\le1$$
• Nested logarithms and greatest-integer parts · Two nested logarithms
  $$\log_a(\log_b u)\ \text{defined}\iff u>1\quad(a,b>1)$$
• Domain of a composite function · Domain of f∘g
  $$\{x\in\operatorname{dom}g:\ g(x)\in\operatorname{dom}f\}$$

Watch out for (3)

• A logarithm in a denominator
  → The conditions that fix a domain
• Turn integers back into intervals
  → Nested logarithms and greatest-integer parts
• The inner function's gaps stay
  → Domain of a composite function

Formulas (2)

• Finding the range · Rational function
  $$y=\frac{p(x)}{q(x)}\ \Rightarrow\ \Delta_x\ge0$$
• Deciding one-one and onto · The two tests
  $$\text{one-one: } f(a)=f(b)\Rightarrow a=b;\qquad \text{onto: } f(A)=B$$

Watch out for (2)

• When the x² term vanishes
  → Finding the range
• Read the codomain
  → Deciding one-one and onto

Formulas (2)

• Counting all functions and one-one functions · One-one functions
  $$\frac{n!}{(n-m)!}=n(n-1)\cdots(n-m+1)$$
• Counting onto functions · Onto functions
  $$n^m-\binom n1(n-1)^m+\binom n2(n-2)^m-\cdots$$

Watch out for (2)

• More inputs than outputs
  → Counting all functions and one-one functions
• Add back the double-counted
  → Counting onto functions

Formulas (3)

• Composing functions, and recovering one from the other · Composition
  $$(f\circ g)(x)=f\big(g(x)\big)$$
• Inverse functions and self-inverse maps · Self-inverse Möbius map
  $$f(x)=\frac{ax+b}{cx+d},\quad f\circ f=\mathrm{id}\iff a+d=0$$
• Applying a function many times · A cycle
  $$f^k=\mathrm{id}\ \Rightarrow\ f^{n}=f^{\,n \bmod k}$$

Watch out for (3)

• The inner function acts first
  → Composing functions, and recovering one from the other
• Only for increasing functions
  → Inverse functions and self-inverse maps
• Find the cycle on a general x
  → Applying a function many times

Formulas (3)

• Additive and multiplicative rules · Additive rule
  $$f(x+y)=f(x)+f(y)\ \Rightarrow\ f(n)=n\,f(1)$$
• Substitute again, then solve the pair · Two equations, two unknowns
  $$af(x)+bf\big(g(x)\big)=h(x),\quad af\big(g(x)\big)+bf(x)=h\big(g(x)\big)$$
• Pairing terms when f(x) + f(a − x) is constant · The standard pair
  $$f(x)=\frac{b^x}{b^x+\sqrt b}\ \Rightarrow\ f(x)+f(1-x)=1$$

Watch out for (3)

• A constant term changes f(0)
  → Additive and multiplicative rules
• A sum may need no solving
  → Substitute again, then solve the pair
• Count the terms
  → Pairing terms when f(x) + f(a − x) is constant