Relations and Functions
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Reflexive, Symmetric and Transitive Relations
Part of: The Language of Relations
A relation is a collection of pairs from a set to itself. It is reflexive when every element pairs with itself, symmetric when every pair can be flipped, and transitive when chains link up. The empty collection and the full collection are the two extreme cases.
Why it matters. These three tests unlock equivalence relations, and examiners love small sets where one property holds and others fail.
Formulas
- Reflexive needs every (a, a) inside R
- Symmetric flips every pair held by R
- Transitive chains (a, b) and (b, c) into (a, c)
Key words
- Reflexive relation
- : A relation holding every self-pair of its set.
- Symmetric relation
- : A relation holding the flip of every pair it holds.
- Transitive relation
- : A relation chaining linked pairs into a direct pair.
- Empty relation
- : A relation holding no pairs at all.
- Universal relation
- : A relation holding every possible pair of its set.
- Trivial relation
- : Either the empty or the universal relation.
Try it in two minutes
- Write the numbers 1, 2 and 3 on slips with arrows for each pair.
- Ask the child to check self-arrows, flipped arrows and chained arrows.
- Sort three example relations by which tests they pass.
Try at home. Build a relation on {1, 2} that is symmetric but neither reflexive nor transitive, and check each test.
Equivalence Relations and Their Classes
Part of: The Language of Relations
A relation passing all three tests is called equivalence, and it chops its set into non-overlapping classes that together cover everything. Same remainders, same parity and same scores all build classes. Intersections of equivalences stay equivalence, but unions can break chains.
Why it matters. Classes turn messy sets into tidy partitions, and counting them is a favourite hard question.
Formulas
- Equivalence means reflexive plus symmetric plus transitive
- Classes split the set with no overlaps
- Intersection of equivalences stays an equivalence
Key words
- Equivalence relation
- : A relation passing the reflexive, symmetric and transitive tests.
- Equivalence class
- : The full team of mates of one element.
- Partition
- : A split of a set into disjoint teams covering everything.
- Remainder class
- : Integers sharing one remainder on division, like mates of 1 mod 5.
Try it in two minutes
- Write numbers 0 to 9 on slips and sort by remainder on division by 5.
- Check that piles never overlap and none is left out.
- Intersect two groupings and test whether chains survive.
Try at home. Split {1, 2, 3, 4, 5, 6} by even difference and list each class.
One-one, Onto and Bijective Functions
Part of: Special Functions and Their Combinations
A function is one-one when distinct inputs never share an output, and onto when every target in the co-domain is hit. Both together mean bijective. On finite same-size sets the two come together, but infinite sets split them: doubling naturals misses odds, while folding odds and evens together covers everything twice.
Why it matters. Bijective maps unlock inverses, and the finite-versus-infinite twist is a classic trap.
Formulas
- One-one keeps distinct inputs apart in output
- Onto covers the whole co-domain
- Finite same-size maps: one-one and onto come together
Key words
- One-one function
- : A function sending distinct inputs to distinct outputs.
- Onto function
- : A function hitting every element of its co-domain.
- Bijective function
- : A function that is both one-one and onto.
- Many-one function
- : A function sharing one output between distinct inputs.
- Range
- : The set of outputs a function actually hits.
- Co-domain
- : The declared set of possible outputs.
Try it in two minutes
- Draw arrows from inputs to outputs for a small map.
- Ask whether any two arrows land together and whether any target is missed.
- Replay the same checks on a doubling map of naturals.
Try at home. Test f(x) = 5x on naturals for one-one and onto, and name what fails.
Composition of Functions and Inverses
Part of: Special Functions and Their Combinations
Chaining f then g builds gof, and swapping the order usually changes the result. A map with a two-sided undo partner is invertible, which happens exactly for bijective maps. The partner is found by solving for the input, as with 5x + 2 undoing to (y − 2) over 5.
Why it matters. Composition models multi-step processes, and inverses reverse them, from coding to finance.
Formulas
- gof means apply f first, then g
- Invertible matches bijective exactly
- Check inverses by composing both ways to identity
Key words
- Composition of functions
- : Chaining two maps so the output of the first feeds the second.
- Invertible function
- : A function owning a partner that undoes it both ways.
- Inverse function
- : The partner map that reverses an invertible function.
- Identity function
- : A map returning every input unchanged.
Try it in two minutes
- Pick an input and walk it through f then g on paper.
- Swap the order and compare the two landings.
- Solve y = 5x + 2 for x to expose the inverse.
Try at home. Compute gof(2) and fog(2) for f doubling and g adding 5, and compare.
What your child will practise
- Composition of Functions and Inverses: Candy Quest (Flip and Match)
- Composition of Functions and Inverses: Jungle Quick Fire (Quick Fire)
- Equivalence Relations and Their Classes: Space Puzzle (Clue Puzzle)
- One-one, Onto and Bijective Functions: Desert Quick Fire (Quick Fire)
- Reflexive, Symmetric and Transitive Relations: Jungle Puzzle (Clue Puzzle)
- Quiz: Composition of Functions and Inverses (8 questions)
- Quiz: Equivalence Relations and Their Classes (8 questions)
- Quiz: One-one, Onto and Bijective Functions (8 questions)
- Quiz: Reflexive, Symmetric and Transitive Relations (8 questions)