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The proof by contradiction method, a powerful logical tool used to disprove statements. By assuming the statement is false and deriving a contradiction, we can establish its truth. This technique is not limited to proving conditional statements and can be applied to various mathematical propositions. examples and exercises to help understand this proof method.
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Typology: Exercises
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P C ∼ P C ∧ ∼ C ( ∼ P ) ⇒ ( C ∧ ∼ C ) T T F F T T F F F T F T T F F F F T F F
( p 1 p 2 p 3 · · · p (^) k − 1 p (^) k p (^) k + 1 · · · p (^) n ) + 1 = c p (^) k.
( p 1 p 2 p 3 · · · p (^) k − 1 p (^) k + 1 · · · p (^) n ) +
p (^) k^ =^ c,
p (^) k
= c − ( p 1 p 2 p 3 · · · p (^) k − 1 p (^) k + 1 · · · p (^) n ).
2 = r
d c.
2 = r
d c =^
a b
d c =^
ad bc.
2 · r /
A. Use the method of proof by contradiction to prove the following statements. (In each case you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.)
1. Suppose n ∈ Z. If n is odd, then n 2 is odd. 2. Suppose n ∈ Z. If n 2 is odd, then n is odd. 3. Prove that 3
( 2 is irrational.
4. Prove that
( 6 is irrational.
5. Prove that
( 3 is irrational.
6. If a, b ∈ Z, then a 2 − 4 b − 2 #= 0. 7. If a, b ∈ Z, then a 2 − 4 b − 3 #= 0. 8. Suppose a, b, c ∈ Z. If a 2 + b 2 = c 2 , then a or b is even. 9. Suppose a, b ∈ R. If a is rational and ab is irrational, then b is irrational. 10. There exist no integers a and b for which 21 a + 30 b = 1. 11. There exist no integers a and b for which 18 a + 6 b = 1. 12. For every positive rational number x , there is a positive rational number y for which y < x. 13. For every x ∈ [ π /2, π ], sin x − cos x ≥ 1. 14. If A and B are sets, then A ∩ ( B − A ) = .. 15. If b ∈ Z and b! k for every k ∈ N, then b = 0. 16. If a and b are positive real numbers, then a + b ≤ 2
( ab.
17. For every n ∈ Z , 4 #| ( n 2 + 2). 18. Suppose a, b ∈ Z. If 4 | ( a 2 + b 2 ), then a and b are not both odd.
B. Prove the following statements using any method from chapters 4, 5 or 6.
19. The product of any five consecutive integers is divisible by 120. (For example, the product of 3,4,5,6 and 7 is 2520, and 2520 = 120 · 21 .) 20. We say that a point P = ( x, y ) in the Cartesian plane is rational if both x and y are rational. More precisely, P is rational if P = ( x, y) ∈ Q 2. An equation F ( x, y) = 0 is said to have a rational point if there exists x 0 , y 0 ∈ Q such that F ( x 0 , y 0 ) = 0. For example, the curve x 2 + y 2 − 1 = 0 has rational point ( x 0 , y 0 ) = (1 , 0). Show that the curve x 2 + y^2 − 3 = 0 has no rational points. 21. Exercise 20 involved showing that there are no rational points on the curve x 2 + y 2 − 3 = 0. Use this fact to show that
( 3 is irrational.
22. Explain why x 2 + y^2 − 3 = 0 not having any rational solutions (Exercise 20) implies x 2 + y 2 − 3 k^ = 0 has no rational solutions for k an odd, positive integer. 23. Use the above result to prove that
√ 3 k^ is irrational for all odd, positive k.