First taste of proofs

This week’s lectures were fairly interesting. I learned when A bears a relation to B and B bears the same relation to C, then A bears it to C. This means that the relationship is transitive. For example, if ∀x∈X, (P(x)⇒Q(X))∧ (Q(x)⇒R(X)) is true then ∀x∈X, P(x)⇒R(X) is also true. Also, the order of quantifiers matters, ∀x∈X, ∃y∈X means that you choose a y after your enemy chooses a x; on the other hand, ∃y∈X, ∀x∈X means that you choose a y and then your enemy chooses a x. The meaning of statements entirely change when the order of the quantifiers are altered. I was also introduced to the definition of a limit during this week’s lecture. I learned how to find a range for a delta when epsilon is given. For example, when given the ∀e∈ℝ⁺, ∃d∈ℝ⁺, ∀x∈ℝ, |x-2|<d⇒|x²-4|<e and e=2 the method for calculating d would be to plug the y values into the antecedent. Which would give the values |2^0.5-2|=0.586 and|6^0.5-2|=0.449 so delta must be less than the smallest difference (d<0.449). The outline for a proof is also pretty interesting. You “assume” when the quantifier is ∀ and “let” when the quantifier is ∃. Also, when the statement consists of an implication you assume the antecedent. After the body of the proof is complete you introduce the quantifiers. What surprised me was the way to find proofs. The two ways introduced in the lecture was bubble search and tree search. In bubble search and tree search you try to find the chain of antecedents and consequents.