Question 1: Define a relation ∼ on R2 by stating that (a, b) ∼ (c, d) if and only if a2 + b2 ≤ c2 + d2. Show that ∼ is reflexive and transitive but not symmetric.
Question 2: Show that an m x n matrix gives rise to a well-defined map from Rn to Rm.
Question 3: Find the error in the following argument by providing a counter example. "The reflexive property is redundant in the axioms for an equivalence relation. If x ∼ y, then y ∼ x by the symmetric property. Using the transitive property, we can deduce that x ∼ x."
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