Relations and Functions
Class 11 · Mathematics
4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.
Ordered Pairs and Cartesian Products
Part of: Pairs, Relations and Functions
Order matters in (a, b) so equality needs both coordinates equal; P × Q pairs every p with every q, counts multiply, null kills the product, and A × A, A × B × C extend naturally.
Why it matters. Pairs encode links between sets. Product counts power every relation.
Formulas
- n(A x B) = n(A) x n(B)
- (a, b) = (c, d) exactly when a = c and b = d
- A x B is empty if A is empty or B is empty
Key words
- Ordered pair
- : Two elements written as (a, b) where order decides equality.
- Cartesian product
- : The set of all (a, b) pairs formed by taking first entries from A and second from B.
- Equality of pairs
- : The rule that (a, b) equals (c, d) exactly when a = c and b = d.
- Empty product
- : The result that A x B is empty whenever A or B is empty.
Try it in two minutes
- Match coordinates in order.
- Pair every p with every q.
- Multiply counts.
Try at home. Pair 3 tops with 4 bottoms.
Relations: Domain, Codomain and Range
Part of: Pairs, Relations and Functions
Relations subset P × Q; domain gathers first entries, range the used seconds, codomain the whole target; arrows, rosters, rules and graphs describe them while total relations count two-to-product-size.
Why it matters. Domain-range reading precedes functions. Relation counts measure all links.
Formulas
- Relations from A to B = 2^(n(A) x n(B))
- Range is a subset of the codomain
Key words
- Relation
- : A subset of A x B linking first entries to second entries.
- Domain
- : The set of all first entries used by a relation.
- Codomain
- : The whole second set named in the relation, whether used or not.
- Range
- : The set of second entries actually used by the relation.
Try it in two minutes
- Check subset of product.
- Collect firsts and seconds.
- Count two-to-product.
Try at home. Relate friends to games they play.
Functions and Special Functions
Part of: Functions and Their Algebra
Functions assign each domain member exactly one image; one-to-many fails the test; identity returns input, constant stays flat, polynomial and rational follow formulas while modulus, signum and greatest-integer step outputs.
Why it matters. Uniqueness of images defines functions. Special types model real rules.
Formulas
- Identity: f(x) = x for all real x
- Constant: f(x) = c for all real x
- Modulus: f(x) = |x| = x if x >= 0, -x if x <= 0
Key words
- Function
- : A relation giving each domain member exactly one image.
- Identity function
- : The function f(x) = x that returns every input unchanged.
- Constant function
- : The function f(x) = c giving the same output for every input.
- Modulus function
- : The function f(x) = |x| giving the non-negative size of x.
- Signum function
- : The function giving 1 above 0, 0 at 0 and -1 below 0.
- Greatest integer function
- : The function f(x) = [x] giving the largest integer at or below x.
Try it in two minutes
- Check one image each.
- Evaluate at the point.
- Classify the type.
Try at home. Test if sibling-of is a function.
Algebra of Real Functions
Part of: Functions and Their Algebra
Sums, differences, products and quotients act pointwise while domains intersect and quotients exclude zero denominators; graphs shift, stretch and reflect these rules.
Why it matters. Pointwise algebra builds models. Domain care avoids hidden division by zero.
Formulas
- (f + g)(x) = f(x) + g(x)
- (f - g)(x) = f(x) - g(x)
- (f . g)(x) = f(x) . g(x)
- (f / g)(x) = f(x) / g(x) with g(x) not 0
Key words
- Sum of functions
- : The sum (f + g)(x) = f(x) + g(x) on the shared domain.
- Product of functions
- : The product (f . g)(x) = f(x) . g(x) on the shared domain.
- Quotient of functions
- : The quotient (f / g)(x) = f(x) / g(x) where g(x) is not zero.
- Domain of combination
- : The overlap of the two domains, minus zeros of g for quotients.
Try it in two minutes
- Combine at each x.
- Intersect domains.
- Exclude zero denominators.
Try at home. Add two price rules pointwise.
What your child will practise
- Algebra of Real Functions: Jungle Puzzle (Clue Puzzle)
- Algebra of Real Functions: Jungle Quick Fire (Quick Fire)
- Functions and Special Functions: Candy Puzzle (Clue Puzzle)
- Ordered Pairs and Cartesian Products: Space Quest (Flip and Match)
- Relations: Domain, Codomain and Range: Space Quick Fire (Quick Fire)
- Quiz: Algebra of Real Functions (8 questions)
- Quiz: Functions and Special Functions (8 questions)
- Quiz: Ordered Pairs and Cartesian Products (8 questions)
- Quiz: Relations: Domain, Codomain and Range (8 questions)