Proof Methods and Strategy - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CSE115/ENGR160 Discrete Mathematics 02/02/

1.8 Proof methods and strategy

• Proof by cases : pi→q for i=1,2,…,n
• When it is not possible to consider all cases at the same time
• Exhaustive proof : some theorems can be proved by examining a relatively small number of examples

2

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(( ) ) 1 2

1 2 p q p q p q

p p p q n

n ↔ → ∧ → ∧ ∧ →

∨ ∨ ∨ → 



• An integer is a perfect power if it equals na, where a is an integer greater than 1
• Prove that the only consecutive positive integers not exceeding 100 that are perfect powers are 8 and 9
• Can prove this fact by examining positive integers n not exceeding 100 - First check whether n is a perfect power, and then check whether n+1 is a perfect power

• For positive integers
  • The squares ≤ 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100
  • The cubes ≤ 100: 1, 8, 27, and 64
  • The 4th^ powers n^4 ≤ 100: 1, 16, and 81
  • The 5th^ powers n^5 ≤ 100: 1 and 32
  • The 6th^ powers n^6 ≤ 100: 1 and 64
  • Look at the list of perfect powers, we see that the pair of n=8 and n+1=9 is the only two consecutive powers ≤ 100 (^5)

• Show that |xy|=|x||y| for real numbers
• x≥0, y≥0: xy ≥0 |xy|=xy=|x||y|
• x≥0, y<0: xy<0 |xy|=-xy=x(-y)=|x||y|
• x<0, y ≥0:xy<0 |xy|=-xy=(-x)y=|x||y|
• x<0, y<0: xy>0 |xy|=xy=(-x)(-y)=|x||y|

7

(( ) ( ) ( ))

(( ) ) 1 2

1 2 p q p q p q

p p p q n

n ↔ → ∧ → ∧ ∧ →

∨ ∨ ∨ → 



• Formulate a conjecture about the decimal digits that occur at the final digit of the squares of an integer and prove the result
• The smallest perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 and so on
• Note that the digits that occur at the final digit of a squares are: 0, 1, 4, 5, 6, and 9 (and no 2, 3, 7, and 8)  conjecture

• Case 2: if the final digit of n is 2 or 8, then the final digit of n^2 is 4
• Case 3: if the final digit of n is 3 or 7, then the final digit of n^2 is 9
• Case 4: if the final digit of n is 4 or 6, then the final digit of n^2 is 6
• Case 5: if the final digit of n is 5, then the final digit of n^2 is 5
• Case 6: if the final digit of n is 0, then the final digit of n^2 is 0

• Show that there are no solutions in integers x and y of x^2 +3y^2 =
• x^2 >8 when |x|≥3, and 3y^2 >8 when |y|≥2. The only values for x are -2,-1,0,1,2 and for y are
  • 1, 0, 1
• So, possible values for x 2 are, 0, 1, and 4. The possible values for 3y^2 are 0 and 3
• No pair of x and y can be solution