Day 2: Power Sets, Specification and Functions
Foundations of Pure Mathematics
Power Set
The power set of a set is the collection of all possible subsets of that set, including the empty set and the set itself. For a set S, the power set is denoted as P(S) or 2^S. The size of the power set is always 2^n, where 'n' is the number of elements in the original set. For example, if S = {a, b}, then the power set
Specification
Set-builder notation, or set specification, is a concise way to describe a set by indicating the properties that its members must satisfy. This is typically written in the form {x | P(x)}, where x represents an element and P(x) is a property that x must satisfy. For example, the set of all even numbers can be specified as {x | x is even}. The elements in the resulting set are those for which the condition P(x) is true.
Functions
A function is a mathematical relation between two sets, known as the domain and codomain, that assigns each element in the domain to exactly one element in the codomain. Functions are often represented as f: A → B, where A is the domain, B is the codomain, and f is the function. Functions can be described using various notations, such as f(x) = y, where x is an element of the domain, and y is the corresponding element in the codomain.
Question 1
Solution
Suppose that f is a function on [n]. We can specify the rule for f by listing, in order, the elements f(0), f(1), f(2), and so on until f(n - 1). There are n possibilities for each and no other restrictions. Thus the number of possible rules is n multiplied by itself n times.
Thank you for reading. Stay tuned for Day 3’s email on types of functions, cardinality & counting, Cantor’s Theorem and Russel’s Paradox.
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Barry 🍩