Proof of uniqueness: Practice Questions

Suppose we wish to prove that the additive identity in a ring \(R\) is unique. We assume \(0_1\) and \(0_2\) are both additive identities. By the definition of an identity, which pair of equations correctly demonstrates that \(0_1 = 0_2\)?

Let \(S \subseteq \mathbb{R}\) be a non-empty set that is bounded below. To prove that the infimum (greatest lower bound) of \(S\) is unique, we assume \(L_1\) and \(L_2\) are both infima of \(S\). Which logical argument correctly concludes \(L_1 = L_2\)?

In mathematical proofs, to establish the uniqueness of an element \(x\) in a set \(S\) satisfying a property \(P\), what is the most common logical starting point?

Let \( P(x) \) be a property of an object in a set \( X \). To provide a proof of uniqueness for an element \( a \in X \) satisfying \( P(a) \), which of the following logical procedures is typically followed after establishing that at least one such element exists?

Let \(f: \mathbb{R} \to \mathbb{R}\) be a strictly monotonic function. To prove that the solution to the equation \(f(x) = L\) is unique (given that it exists), which property is utilized when assuming two solutions \(x_1\) and \(x_2\) where \(x_1 \neq x_2\)?

A function is said to be unique if it is the only function satisfying a set of given properties. Consider the differential equation \(\frac{dy}{dx} = y\) with the initial condition \(y(0) = 1\).

(a) Show by direct substitution that the function \(f(x) = e^x\) is a solution to this initial value problem.
(b) Use the proof of uniqueness principle by assuming there exists another solution \(g(x)\). Define a helper function \(h(x) = g(x)e^{-x}\) and show that its derivative is zero for all \(x\).
(c) Conclude that \(g(x) = e^x\) for all \(x\), thereby proving the solution is unique.