Chapter 2: Logic and Incidence Geometry PDF

Republic of the Philippines CENTRAL BICOL STATE UNIVERSITY OF AGRICULTURE Calabanga | Pasacao | Pili | Sipocot CHAPTER 2: LOGIC AND INCIDENCE GEOMETRY Prepared by: ELAIN JOYCE V. PONDALIS  INFORMAL LOGIC Logic Rule 0. No unstated assumptions may be used...

Republic of the Philippines CENTRAL BICOL STATE UNIVERSITY OF AGRICULTURE Calabanga | Pasacao | Pili | Sipocot CHAPTER 2: LOGIC AND INCIDENCE GEOMETRY Prepared by: ELAIN JOYCE V. PONDALIS  INFORMAL LOGIC Logic Rule 0. No unstated assumptions may be used in a proof. Examples: Triangle assumption: If you're proving something about a triangle, you can't just assume it's a right triangle without that being explicitly stated in the problem. All deductions must flow logically from what is given. Equality of numbers: If you're proving properties of two numbers, a and b, you can't assume a=b unless this has been stated or proven in previous steps. Implication proof: When proving P⟹Q, you can't assume Q is true from the start. Instead, you must show it follows from P. THEOREMS AND PROOFS ✓ All mathematical theorems are conditional statements of the form If [hypothesis] then [conclusion]. ✓ Symbolically: 𝐻⟹𝐶, H implies C ✓ A proof is a list of statements, together with the justification for each statement, ending with the desired conclusion. ✓ A theorem is a statement that has a proof. ✓ Other names for theorem: proposition, corollary, lemma Logic Rule 1 The following are the six types of justifications allowed for statements in proofs: (1) “By hypothesis…” (2) “By axiom…” (3) “By theorem…” (previously proved) (4) “By definition…” (5) “By step…” (a previous step in the argument) (6) “By rule… of logic. ” RAA PROOFS Reductio ad absurdum(RAA), which means reduction to the absurd, is a common proof technique. To prove 𝐻⟹𝐶, start by assuming the negation of C. (RAA hypothesis). Show that the negation of C leads to absurdity. Conclude that C must be valid. (RAA conclusion) Proof by Contradiction is an RAA proof. Logic Rule 2 (RAA Proof) To prove 𝐻⟹𝐶, start by assuming the negation of C and deduce an absurd statement, using the hypothesis H if needed in your deduction. RAA Proof Example: 2 is irrational Suppose 2 is rational (RAA hypothesis). Then 2 = 𝑝/𝑞 for some integers 𝑝 and 𝑞. Assume 𝑝/𝑞 is a reduced fraction, 𝑝 and 𝑞 have no common factors. Then 2𝑞 = 𝑝, and squaring both sides gives 2𝑞2 = 𝑝2. Since 𝑝2 is even, 𝑝 must be even. That is, 𝑝 = 2𝑟 for some integer 𝑟. We have 2𝑞 2 = 𝑝2 = 2𝑟 2 = 4𝑟 2. Cancel 2 from both sides. Then 𝑞 2 = 2𝑟 2 , and 𝑞 must be even too! This is absurd because 𝑝 and 𝑞 have no common factors. Thus 2 is irrational (RAA conclusion).∎ RAA Proof Example: If l and m are distinct lines that are not parallel, then l and m have a unique point in common. Because l and m are not parallel, they have a point in common (by definition of “parallel”). Since we want to prove the uniqueness for the point in common, we will assume the contrary, that l and m have two distinct points A and B in common (RAA hypothesis). Then there is more than one line on which A and B both lie A and B lie on a unique line (Euclid’s Postulate 1) Thus, Intersection of l and m is unique (RAA conclusion).∎ NEGATION Notation: S and R are statements. The negation or contrary of S is denoted by ~S. S&R is a conjunction that means “S and R.” S or R is a disjunction that means “S or R or both.” A contradiction is a statement of the form S&~S. Logic Rule 3 The statement “~(~S)” means the same as S. Examples: 1. ∼(∼P)≡P : The negation of "not P" is simply P. 2. ∼(∼(∼Q))≡∼Q: If you negate the negation of Q, you end up with Q. 3. ∼(∼(x>3))≡x>3: The negation of "not x>3" is simply x>3. Logic Rule 4 The statement “~[𝐻 → 𝐶]” means the same as “𝐻&~𝐶.” Examples: The negation of “ If x is prime, then x is odd” is “x is prime and x is even.” ∼(P⟹Q)≡P∧∼Q: If the implication P ⟹ Q is false, then P must be true, and Q must be false. Logic Rule 5 The statement “~(S&R)” means the same as “~S or ~R.” Examples: ~(The natural number x is odd and prime.) The natural number x is even or composite. ∼(P∧Q)≡∼P∨∼Q: The negation of "both P and Q" is "either P is false or Q is false.“ ∼(x>3∧y