Sets, Relations and Functions in Business
Sets, relations and functions are the vocabulary business mathematics uses to describe choices, mappings and economic links. The exam tests set operations, the difference between a relation and a function, domain and range, and the construction of demand, cost, revenue and profit functions from a business situation.
Core rules
- A set is a well-defined collection. Union, intersection, difference and complement are read from Venn diagrams; A × B is the set of ordered pairs (a, b).
- A relation from A to B is any subset of A × B. It is a function only if every input in the domain has exactly one output.
- Standard business models: demand q = f(p), total cost C(q) = fixed cost + variable cost per unit × q, revenue R(q) = p × q, profit π(q) = R(q) − C(q). Break-even means R(q) = C(q), so π(q) = 0.
- Quantity and price are normally restricted to non-negative values, and demand is usually a decreasing function of price.
Worked example
Demand is q = 100 − 2p and total cost is C(q) = 10q + 600. Revenue is R(p) = p(100 − 2p) = 100p − 2p². At p = 30, q = 40, so R = ₹1,200 and C = ₹1,000; profit is ₹200. Break-even needs R(p) = C(q). Substituting gives 100p − 2p² = 10(100 − 2p) + 600 = 1,600 − 20p, so p² − 60p + 800 = 0, i.e. (p − 20)(p − 40) = 0. Break-even prices are ₹20 and ₹40, with quantities 60 and 20. The two roots show that raising price can cut volume enough to remove profit.
Traps
- Calling a relation a function when one input has two outputs.
- Using R = p instead of R = p × q, or mixing price and quantity in the cost function.
- Omitting domain restrictions and then accepting an economically meaningless negative quantity.