CBSE · Class 12 · Mathematics · Chapter 1

Relations and Functions — Formula Sheet

Board Formulas

15 formulas

1.Relation in a Set
$R \subseteq A \times A, a R b \Leftrightarrow (a, b) \in R$
A relation from A to B is any subset of A × B. A relation in A is a subset of A × A.
2.Empty and Universal Relations
$R = ϕ (empty), R = A \times A (universal)$
Empty: no element is related to any element. Universal: every element is related to every element. Both are called trivial relations.
3.Reflexive Relation
$(a, a) \in R \forall a \in A$
Every element of A must be related to itself. A single missing (a, a) makes R non-reflexive.
4.Symmetric Relation
$(a, b) \in R \Rightarrow (b, a) \in R \forall a, b \in A$
Every pair must have its reverse in R. Pairs of the form (a, a) never break symmetry.
5.Transitive Relation
$(a, b) \in R and (b, c) \in R \Rightarrow (a, c) \in R$
Check every chain a → b → c. If no chain exists at all, the relation is transitive by default.
6.Equivalence Relation★
$R is reflexive, symmetric and transitive$
Typical examples: 'is congruent to', 'is parallel to' (with each line parallel to itself), and 'a − b is divisible by n' on Z.
7.Equivalence Class★
$[a] = {b \in A : (a, b) \in R}$
The equivalence classes are pairwise disjoint and their union is the whole set A. Two classes are either equal or have no element in common.
8.Congruence Modulo n on Z
$R = {(a, b) : n ∣ (a - b)}, classes [0], [1], \dots, [n - 1]$
$n$ : Fixed positive integer (the modulus) · $[r]$ : Class of integers with remainder r on division by n
An equivalence relation on the integers with exactly n classes. [r] is the set of integers that leave remainder r on division by n.
9.One-one (Injective) Function★
$f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2} \forall x_{1}, x_{2} \in X$
Distinct inputs give distinct outputs. If two different inputs share an image, f is many-one. A function that is strictly increasing or strictly decreasing on its domain is one-one.
10.Onto (Surjective) Function★
$\forall y \in Y \exists x \in X : f (x) = y \Leftrightarrow Range of f = Y$
Every element of the co-domain must be hit. Whether f is onto depends on the co-domain, not only on the formula.
11.Bijective Function
$f bijective \Leftrightarrow f is one-one and onto$
Prove the two parts separately. Showing only one of them does not establish bijectivity.
12.Functions on a Finite Set
$X finite, f : X \to X : f one-one \Leftrightarrow f onto$
Works only for finite sets mapped to themselves. On N, f(x) = 2x is one-one but not onto.
13.Number of Relations
$∣ A ∣ = m, ∣ B ∣ = n \Rightarrow number of relations from A to B = 2^{mn}$
$m$ : Number of elements in A · $n$ : Number of elements in B
A × B has mn ordered pairs, and each relation is a subset of it. A relation in A alone (m = n) gives 2^(n²).
14.Number of Functions and One-one Functions
$functions A \to B = n^{m}, one-one functions = \frac{n !}{( n - m )!} (m \leq n)$
$m$ : Number of elements in the domain A · $n$ : Number of elements in the co-domain B
Each of the m elements of A has n choices of image. For one-one maps the choices fall as n, n − 1, ..., so no image repeats. If m > n there is no one-one function.
15.Number of Bijections of a Set onto Itself
$∣ A ∣ = n \Rightarrow number of bijections A \to A = n!$
A bijection of a finite set onto itself is just a permutation of its elements. For A = {1, 2, 3} there are 3! = 6.