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MATH 151

MATH 151. Dr. Maryam AL- Towaileb mtowaileb@ksu.edu.sa. Test dates. First Midterm (Sunday ) 2121441 H ( 20 Oct.) 20% Second Midterm (Tuesday) 2231441 H (19 Nov.) 20%. Methods of Proof. Methods of Proof. 3- Contrapositive Proof. 4- By Induction. 2-

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MATH 151

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  1. MATH 151 Dr. Maryam AL-Towaileb mtowaileb@ksu.edu.sa Dr. Halimah Alshehri

  2. Test dates • First Midterm (Sunday ) 21\2\1441 H (20 Oct.) • 20% • Second Midterm (Tuesday) 22\3\1441 H (19 Nov.) • 20% Dr. Maryam AL-Towaileb

  3. Methods of Proof Dr. Maryam AL-Towaileb

  4. Methods of Proof 3- Contrapositive Proof 4- By Induction 2- Indirect Proof (Contradiction) 1- Direct Proof Dr. Maryam AL-Towaileb

  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. Maryam AL-Towaileb

  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. Maryam AL-Towaileb

  7. Example1: Use direct proof to show that : If n is an odd integer then is also an odd integer. Dr. Maryam AL-Towaileb

  8. Dr. Maryam AL-Towaileb

  9. Example2: Use a direct proof to show that the sum of two even integers is even. Dr. Maryam AL-Towaileb

  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. Maryam AL-Towaileb

  11. Example 3: Use a direct proof to show that every odd integer is the difference of two squares. Dr. Maryam AL-Towaileb

  12. 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. Maryam AL-Towaileb

  13. Example 3: • Use indirect proof to show that : If is an odd then so is . Dr. Maryam AL-Towaileb

  14. Dr. Maryam AL-Towaileb

  15. 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. Maryam AL-Towaileb

  16. Example 4: • Let x be an integer. Prove that : If x² is even, then x is even. (by Contrapositive proof) Dr. Maryam AL-Towaileb

  17. Dr. Maryam AL-Towaileb

  18. 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. Maryam AL-Towaileb

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