I need notes for my Cs 330 class on deduction proofs
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QUESTION 7 [30] In this question, you have to construct formal proofs using the natural deduction rules. The Fitch system makes use of these rules. A summary of the rules of natural deduction is given on pages 573 to 578 of your textbook. Consult this when you do this question. Remember that De Morgan’s laws and other tautologies are not permissible natural deduction rules. You are also not allowed to use Taut Con, Ana Con or FO Con. It is important to number your statements, to indicate subproofs and at each step to give the rule that you are using. Hint: If you have access to a computer, take advantage of the fact and use Fitch.
Akash M.
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Reasoning and Proofs
Proving Geometric Relationships
1. 1.2.6 (a) Verify the triangle inequality in the special case where a and b have the same sign. (b) Find an efficient proof for all the cases at once by first demonstrating (a + b)^2 ≤ (|a| + |b|)^2. (c) Prove |a - b| ≤ |a - c| + |c - d| + |d - b| for all a, b, c, d. (d) Prove ||a| - |b|| ≤ |a - b|. (The unremarkable identity a = a - b + b may be useful.) 2. Let x, y ∈ R. Prove the following properties using only the ordered field axioms: If x ≠ 0 and xy = xz, then y = z. 3. Prove that (2Z + 1) is countable. 4. 1.5.3 Use the following outline to supply proofs for the statements in Theorem 1.5.8. (a) Use induction to prove Theorem 1.5.8 (i). You may follow the following outline: First, prove statement (i) for two countable sets, A1 and A2. Example 1.5.3 (ii) may be a useful reference. Some technicalities can be avoided by first replacing A2 with the set B2 = A2 A1 = {x ∈ A2 : x ∉ A1}. The point of this is that the union A1 ∪ B2 is equal to A1 ∪ A2 and the sets A1 and B2 are disjoint. (What happens if B2 is finite?) Next, prove the more general statement in (i). (b) Explain why induction cannot be used to prove part (ii) of Theorem 1.5.8 from part (i). (c) Explain/show how arranging N into the two-dimensional array 1 3 6 10 15 ... 2 5 9 14 ... 4 8 13 ... 7 12 ... 11 ... leads to a proof of Theorem 1.5.8 (ii). 5. 1.5.8 Let B be a set of positive real numbers with the property that adding together any finite subsets of elements from B always gives a sum of 2 or less. Prove that B must be finite or countable. (Suggestion: Argue that B is a countable union of sets Bn defined for each n ∈ N as the intersection of B and (2/(n+1), 2/n]. For each n, find a (finite) bound on the cardinality of Bn. What can we say about the countability of this union?)
Sri K.
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