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Proof Methods
Some basic questions on proofs using different methods of prove
27
Computer Science
Undergraduate 1
03/12/2012

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Prove the square root of 2 is an irrational number.
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Prove if a divides b and b divides c then a divides c.
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Prove Every odd integer is the difference of two perfect squares.
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Prove The number 100...01 (with 3n-1 zeros where n is an integer larger then 0) is composite.
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Prove If r1 and r2 are distinct roots of the polynomial p(x) = x2 + b x + c, then r1 + r2 = - b and r1 r2 = c.
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Prove If a divides b and a divides c then a divides b + c. (Here a, b, and c are positive natural numbers and the definition of divisibility is given above.)
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Prove If a is an integer, divisible by 4, then a is the difference of two perfect squares.
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Prove If a and b are real numbers, then a2 + b2 >= 2 a b.
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Prove The sum of two rational numbers is a rational number.
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Prove If r1, r2, r3 are three distinct (no two the same) roots of the polynomial p(x) = x3 + b x2 + c x+ d, then r1 r2 + r1 r3 + r2 r3 = c.
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Prove There are no positive integer solutions to the diophantine equation x2 - y2 = 1.
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Prove There are no rational number solutions to the equation x3 + x + 1 = 0.
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Prove There are no rational number solutions to the equation x3 + x + 1 = 0.
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There are no positive integer solutions to the diophantine equation x2 - y2 = 10.
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There is no rational number solution to the equation x5 + x4 + x3+x2+ 1 = 0.
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If a is a rational number and b is an irrational number, then a+b is an irrational number.
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If x and y are two integers for which x+y is even, then x and y have the same parity. (Two integers are said to have the same parity if they are both odd or both even.)
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If x and y are two integers whose product is even, then at least one of the two must be even.
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If x and y are two integers whose product is odd, then both must be odd.
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If a and b a real numbers such that the product a b is an irrational number, then either a or b must be an irration number.
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If a is an integer, then a is not evenly divisible by 3 if, and only if, a2 -1 is evenly divisble by 3.
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A postive integer n is evenly divisible by 3 if, and only if, the sum of the digits of n is divisble by 3.
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If a is an integer, then a is not evenly divisible by 5 if, and only if, a4 -1 is evenly divisble by 5.
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For two integers a and b, a+b is odd if, and only if, exactly one of the integers, a or b, is odd.
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For two integers a and b, the product ab is even if and only if at least one of the integers, a or b, is even.
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A postive integer n is evenly divisible by 9 if, and only if, the sum of the digits of n is divisble by 9.
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A postive integer n is evenly divisible by 11 if, and only if, the difference of the sums of the digits in the even and odd positions in n is divisible by 11.
Definition
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