Problems

The set of symmetries of an object (e.g. a square) is called a group. We can formally define a group \(G\) as follows.

A is a non-empty set \(G\) with a binary operation \(*:G\times G\to G\) satisfying the following axioms (you can think of them as rules that \(G\) and \(*\) have to satisfy). A binary operation takes two elements of \(G\) and gives another element of \(G\).

1. Associativity: For all \(g\), \(h\) and \(k\) in \(G\), \((g*h)*k=g*(h*k)\).

2. Identity: There is an element \(e\) of \(G\) such that \(e*g=g=g*e\) for all \(g\) in \(G\).

3. Inverses: For every \(g\) in \(G\), there exists a \(g^{-1}\) in \(G\) such that \(g*g^{-1}=e\).

4. Closure: For all \(g\) and \(h\) in \(G\), \(g*h\) is also in \(G\).

Prove that the symmetries of the ‘reduce-reuse-recycle’ symbol form a group.

Let \(X\) be a finite set, and let \(\mathcal{P}X\) be the power set of \(X\) - that is, the set of subsets of \(X\). For subsets \(A\) and \(B\) of \(X\), define \(A*B\) as the symmetric difference of \(A\) and \(B\) - that is, those elements that are in either \(A\) or \(B\), but not both. In formal set theory notation, this is \(A*B=(A\cup B)\backslash(A\cap B)\).

Prove that \((\mathcal{P}X,*)\) forms a group.