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Use a proof by contraposition to show that if x + y ≥ 2, where x and y are real numbers, then x ≥ 1 or y ≥ 1. Solution:----- 2. Prove that if m and n are integers and mn is even, then m is even or n is even. Solution:
Procedure \(\PageIndex{1}\): Proof by proving the contrapositive. To prove \(P \Rightarrow Q\text{,}\) you can instead prove \(\neg Q \Rightarrow \neg P\text{.}\)
Proposition. If P, then Q. Proof. Suppose » Q. ... Therefore » P. ç. ositive proof is very simple. The first line of the proof is the sentence “Suppose Q is not true.” . Or something to that effect.) The last line is the sentence . Therefore P is not true.” Between the first and last line we use logic and definitions to transform the stateme.
The converse is used primarily in three places. First, when a mathematician proves a major theorem, often the next step is to explore if the converse is true, as well as why or why not. Second, a major error in proof writing is when a student proves \if Q then P" when the instructor asks for a proof of \if P then Q."
Exercise 16.1 Use the following examples to practise proof by contrapositive. Consider why this method is easier than a direct proof for these conjectures. Conjecture 16.1 : If \ (a^2+b^2\) is odd and \ (a\) and \ (b\) are both integers, then \ (a\) or \ (b\) have different parity to one another.
Proof by contrapositive. Start of proof: Let \(a\) and \(b\) be integers. Assume that \(a\) and \(b\) are even. End of proof: Therefore \(a^2 + b^2\) is even.
Solve the following problems. All proofs must be written according to conventions for formal proofs, including typeface rules (e.g., italic variables, emphasized labels for theorems and proofs, etc.); pay particular attention to the relatively new guideline of stating the method that a proof uses early in that proof. Question 1.
