Reflexive, symmetric, transitive and equivalence relations; one-one (injective), onto (surjective) and bijective functions. Our own step-by-step solutions to Exercises 1.1, 1.2 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. Reflexive: \((a, a) \in R\) for every \(a\). Symmetric: \((a, b) \in R \Rightarrow (b, a) \in R\). Transitive: \((a, b), (b, c) \in R \Rightarrow (a, c) \in R\). All three: an equivalence relation, which splits the set into disjoint classes. One counterexample is enough to show a property fails.
What it tests. One-one: \(f(x_1) = f(x_2) \Rightarrow x_1 = x_2\) (to disprove, find two inputs with the same output). Onto: every element of the codomain is an output (solve \(f(x) = y\) for \(x\) in the domain). Bijective: both.
What it tests. Mixed practice: prove one-one by showing \(f(x_1) = f(x_2) \Rightarrow x_1 = x_2\) (split into cases where the rule changes), onto by solving \(f(x) = y\) inside the domain; count relations or functions on a small set by listing them.
Relations and Functions has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Relations and Functions · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.