📚 Key Concepts
Relations- A relation R from A to B is a subset of A × B (Cartesian product)
- Domain: Set of first elements in ordered pairs
- Range: Set of second elements in ordered pairs
- Can be represented through set notation, arrow diagrams, graphs
Types of Relations- Reflexive: (a,a) ∈ R for all a ∈ A
- Symmetric: If (a,b) ∈ R then (b,a) ∈ R
- Transitive: If (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R
- Equivalence: Relation that is reflexive, symmetric and transitive
Functions- Special type of relation where each element in domain has exactly one image
- One-to-one (Injective): Each element in codomain has at most one pre-image
- Onto (Surjective): Each element in codomain has at least one pre-image
- Bijective: Both one-to-one and onto
🎯 Key Formulas
Domain of f∘g = {x ∈ Domain of g: g(x) ∈ Domain of f}(f∘g)(x) = f(g(x))For inverse function: f⁻¹(f(x)) = x and f(f⁻¹(x)) = x
⚠️ Common Mistakes to Avoid
Confusing domain and rangeIncorrectly identifying function typesWrong composition orderAssuming every function has an inverseMixing up relation and function properties
📖 Knowledge Prerequisites
Set theory basicsCartesian productsBasic coordinate geometryUnderstanding of mappings
💡 Tips for Students
Draw arrow diagrams to visualize relations/functionsCheck all conditions systematically for relation typesFor function composition, work from inside outUse vertical line test for functions in graphsRemember: not all functions have inverses
👉 Practice Recommedations
Identify domains and rangesTest relations for different propertiesConvert between different function representationsFind composite functionsDetermine if functions are invertibleGraph basic functions and their inverses
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