The fourteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers set-valued functions. Images and preimages (inverse images) of sets under functions: definitions, notation and example. Definition and examples of composition of functions. Associativity of function composition.
The twelfth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers notation for recurring decimal expansions. Real numbers with more than one decimal expansion. Real numbers have recurring/terminating decimal expansions if and only if they are rational.