Why Relations and Functions matter in ISC Class 12?
Ever wondered how a recipe turns a list of ingredients into a tasty dish? In math, that “recipe” is a function, and the “ingredients‑to‑dish” pairing is called a relation. Mastering them is the key to cracking many ISC board questions.
💡 In Simple Words: A relation is just a set of ordered pairs, like a match‑making list. A function is a special relation where each first element (the input) points to only one second element (the output). Think of a vending machine: you press one button (input) and get one snack (output).
What is a Relation?
Understanding a Relation
A relation between two sets A and B is any collection of ordered pairs (a, b) where a ∈ A and b ∈ B. The pair tells us “a is related to b”. No extra rules are needed – a can appear many times, and a single b can be paired with many a’s.
Example: Let A = {1,2,3} and B = {a,b}. The set R = {(1,a),(2,a),(2,b)} is a relation from A to B.
Functions: The Special Kind of Relation
Definition of a Function
A function f from set A to set B (written f: A → B) is a relation with one extra condition: each element of A appears as the first component of exactly one ordered pair. In other words, every “input” gets a single “output”.
Think of a school locker system – each student (input) gets one unique locker number (output). No student shares two lockers, and no locker is assigned to two different students.
Key Properties of Relations and Functions
Common Types of Relations
| Property | Meaning | Example |
|---|---|---|
| Reflexive | Every element relates to itself (a, a) for all a∈A | {(1,1),(2,2)} |
| Symmetric | If (a, b) is in R then (b, a) is also in R | {(1,2),(2,1)} |
| Transitive | If (a, b) and (b, c) are in R, then (a, c) must be in R | {(1,2),(2,3),(1,3)} |
| Equivalence | Relation that is reflexive, symmetric, and transitive together | “Same remainder when divided by 3” on integers |
| Function | Each a∈A appears exactly once as first component | {(1,4),(2,5),(3,6)} |
Worked Example 1 – Is a Given Set a Relation?
Let A = {0,1,2} and B = {0,1,2,3}. Check whether R = {(0,1),(1,2),(2,3)} is a relation from A to B.
Step 1: Verify each pair uses an element from A as the first component and an element from B as the second.
- (0,1) – 0∈A, 1∈B ✅
- (1,2) – 1∈A, 2∈B ✅
- (2,3) – 2∈A, 3∈B ✅
All pairs satisfy the condition, so R is indeed a relation from A to B.
Worked Example 2 – Is a Set a Function?
Consider f = {(−2,4),(0,0),(2,4)} from ℝ to ℝ. Determine if f is a function.
Look at the first components: −2, 0, 2 – each appears only once. No input repeats with a different output, so the rule “one input, one output” holds. Hence f is a function.
Domain, Range, and Codomain
The domain is the set of all possible inputs (the first components). The range is the set of actual outputs that appear (the second components actually used). The codomain is the larger set that the function is declared to map into, even if some elements never appear.
In the previous example, domain = {−2,0,2}, range = {0,4}, and if we said f: ℝ → ℝ, the codomain is all real numbers.
Quick Revision Summary
- Relation = any set of ordered pairs between two sets.
- Function = relation with “one‑to‑one” rule for inputs.
- Check reflexive, symmetric, transitive to classify relations.
- Domain = all first elements; range = all second elements actually used.
- Remember the vending‑machine analogy – one button, one snack.
📝 Likely Exam Questions
1. Define a function and give an example that is not a function.
Model answer: A function f: A→B assigns exactly one element of B to each element of A. Example of a relation that is not a function: {(1,2),(1,3)} – the input 1 has two different outputs.
2. State and explain the three properties that make a relation an equivalence relation.
Model answer: Reflexive (a relates to itself), symmetric (if a relates to b then b relates to a), transitive (if a relates to b and b relates to c, then a relates to c). When all three hold, the relation partitions the set into equivalent classes.
3. Given the relation R = {(x,y) | y = x², x ∈ ℤ, 0 ≤ x ≤ 3}, find its domain and range.
Model answer: Domain = {0,1,2,3}. Range = {0,1,4,9} because we square each domain element.
4. Determine whether the relation S = {(a,b),(b,c),(a,c)} on set {a,b,c} is transitive.
Model answer: Yes, because whenever (x,y) and (y,z) are in S, (x,z) is also in S. Here (a,b) and (b,c) imply (a,c), which is present.
5. Explain the difference between range and codomain with a short example.
Model answer: Codomain is the set declared as the target of the function; range is the set of actual outputs. For f: ℝ→ℝ defined by f(x)=x², codomain = ℝ, but range = [0,∞) because squares are never negative.