Relations and Functions Class 12 Notes

Types of Relations

• Reflexive: a R a for every
• $a \in A$
• .
• Symmetric: a R b implies b R a.
• Transitive: a R b and b R c imply a R c.
• An equivalence relation is reflexive, symmetric and transitive — e.g. being in the same class among students.

Types of Functions

• One-one (injective):
• $f(a) = f(b) \Rightarrow a = b$
• .
• Onto (surjective): every element of B is the image of some element of A, i.e. range = codomain.
• Bijective: both one-one and onto; only bijective functions are invertible.
• Number of functions from A to B:
• $n(B)^{n(A)}$
• .

Composition and Invertible Functions

• A function is invertible if and only if it is bijective.
• The inverse satisfies
• $f^{-1} \circ f = I_A$
• and
• $f \circ f^{-1} = I_B$
• .
• To find the inverse: solve y = f(x) for x and swap the roles of x and y.

Binary Operations

• A binary operation * on a set A maps each ordered pair (a, b) to an element of A.
• Commutative:
• $a * b = b * a$
• .
• Associative:
• $(a * b) * c = a * (b * c)$
• .
• Identity element e satisfies
• $a * e = e * a = a$
• ; the inverse of a is the element that gives the identity.

Solved Examples

Example: Show that the function $f(x) = 2x + 3$ on the reals is invertible, and find $f^{-1}(x)$.

Solution: For one-one, $f(a) = f(b) \Rightarrow 2a + 3 = 2b + 3 \Rightarrow a = b$. It is onto since every real y is hit. So f is bijective and invertible. Let $y = 2x + 3$, giving $x = (y - 3)/2$, hence $f^{-1}(x) = \frac{x-3}{2}$.