Sets, Functions and Relations
Sets
Know the language: element, subset, universal set (xi), complement (A'), union (A union B), intersection (A intersect B), empty set and set-builder notation. A set with n elements has 2^n subsets. For two sets, n(A union B) = n(A) + n(B) - n(A intersect B); this formula, not guesswork, solves most Venn-diagram word problems. Practise shading regions on two- and three-set Venn diagrams and translating only A, A but not B and neither into symbols.
Functions and relations
A relation is any set of ordered pairs; a function is a relation in which each input has exactly one output. Understand domain (allowed inputs), range (actual outputs) and codomain, and read these from graphs, tables, arrow diagrams and mappings. Classify mappings as one-to-one, many-to-one, one-to-many or many-to-many (only the first two are functions of x). Use function notation confidently: f(3) means substitute 3; fg(x) means f(g(x)), applied right to left. Find inverse functions by swapping x and y and rearranging, remembering that the domain of the inverse is the range of the original.
Traps
- Only versus total: if 7 study both and 20 study Maths, then 13 study Maths only.
- The empty set and {0} are different; {0} has one element.
- Order matters in composite functions: fg usually differs from gf.
- An inverse function exists only for a one-to-one mapping.
Rule of thumb
Fill the intersection first, then work outwards to the only regions, and finish with the universal set.
Example: 35 students; 20 take Maths, 18 take Physics, 7 take both. Neither = 35 - (13 + 7 + 11) = 4.