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Methods of Proof in Mathematics by Dr. Halimah Alshehri

Learn different methods of proof in mathematics including direct, contrapositive, and indirect proofs with examples shown by Dr. Halimah Alshehri. Test dates are provided.

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Methods of Proof in Mathematics by Dr. Halimah Alshehri

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  1. MATH 151 Dr. Halimah Alshehri haalshehri@ksu.edu.sa Dr. Halimah Alshehri

  2. Test dates • First Midterm (Wednesday ) 15\6\1440 H (20 Feb.) • Second Midterm (Wednesday) 5\8\1440 H (10 Apr.) • Final (Sunday) 16\8\1440 H (21 Apr.) Dr. Halimah Alshehri

  3. Methods of Proof Dr. Halimah Alshehri

  4. Methods of Proof 3- Contrapositive Proof 4- By Induction 2- Indirect Proof (Contradiction) 1- Direct Proof Dr. Halimah Alshehri

  5. DEFINITION: • 1. An integer number n is evenif and only if there exists a number k such that n = 2k. • 2. An integer number n is odd if and only if there exists a number k such that n = 2k + 1. Dr. Halimah Alshehri

  6. Direct Proof: The simplest and easiest method of proof available to us. There are only two steps to a direct proof: 1. Assume that P is true. 2. Use P to show that Q must be true. Dr. Halimah Alshehri

  7. Example1: Use direct proof to show that : If n is an odd integer then is also an odd integer. Dr. Halimah Alshehri

  8. Dr. Halimah Alshehri

  9. Example2: Use a direct proof to show that the sum of two even integers is even. Dr. Halimah Alshehri

  10. Use a direct proof to show that the sum of two even integers is even. • Suppose that m and n are two even integers , so ∃ 𝑘,𝑗 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑚=2𝑘 𝑎𝑛𝑑 𝑛=2j. • Then 𝑚+𝑛=(2𝑘)+(2𝑗) • =2k+2j • =2(𝑘+𝑗) • =2𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑡=𝑘+𝑗. • Thus, 𝑚+𝑛 is even. □ Dr. Halimah Alshehri

  11. Example 3: Use a direct proof to show that every odd integer is the difference of two squares. Dr. Halimah Alshehri

  12. Dr. Halimah Alshehri

  13. Indirect proof (Proof by Contradiction) • The proof by contradiction is grounded in the fact that any proposition must be either true or false, but not both true and false at the same time. • 1. Assume that P is true. • 2. Assume that is true. • 3. Use P and to demonstrate a contradiction. Dr. Halimah Alshehri

  14. Example 3: • Use indirect proof to show that : If is an odd then so is . Dr. Halimah Alshehri

  15. Dr. Halimah Alshehri

  16. Proof by Contrapositive Recall that first-order logic shows that the statement P ⇒ Qis equivalent to ¬Q ⇒ ¬P. • 1. Assume ¬Q is true. • 2. Show that ¬P must be true. • 3. Observe that P ⇒ Q by contraposition. Dr . Halimah Alshehri

  17. Example 4: • Let x be an integer. Prove that : If x² is even, then x is even. (by Contrapositive proof) Dr. Halimah Alshehri

  18. Dr. Halimah Alshehri

  19. Prove that: For all integers m and n, if m and n are odd integers, then m + n is an even integer. (using direct proof) • Show that by (Contradiction): For x is an integer. If 3x+2 is even, then x is even. • Using (Proof by Contrapositive) to show that: For x is an integer. If 7x+9 is even, then x is odd. Dr, Halimah Alshehri

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