Definitions. Proof by Contradiction We now introduce a third method of proof, called proof by contra-diction. This means that if is even then x is even.
The contrapositive of this statement is: “if a² ≠ b² + c² then the triangle in not right-angled at ‘A’”. MAT231 (Transition to Higher Math) Proof by Contradiction Fall 2014 3 / 12. If you can prove that the contrapositive of a statement is true then the original statement must also be true..
Then which is an odd number. So, 0 = (x + y) (x y) = 2y. This new method is not limited to proving just conditional statements – it can be used to prove any kind of statement whatsoever. Though the proofs are of equal length, you may feel that the con-trapositive proof ﬂowed more smoothly. Proof: Suppose x;y 2N and x2 y2 = 1. A First Example: Proof by Contradiction Proposition: There are no natural number solutions to the equation x2 y2 = 1. QED. Example: Parity Here is a simple example that illustrates the method.
Show that this is Show that this is not an if and only if statement by giving a counterexample to the converse. For example, the assertion "If it is my car, then it is red" is equivalent to "If that car is not red, then it is not mine". Then (x y) = (x + y) = 1.
So, to prove "If P, Then Q" by the method of contrapositive means to prove "If Not Q, Then Not P". If $n$ is even then $n$ is not prime. The basic idea is to assume that the statement we want to prove is false, and then show that this assumption leads to nonsense. Proof by Contrapositive July 12, 2012 So far we’ve practiced some di erent techniques for writing proofs. The method of contradiction is an example of an indirect proof: one tries to skirt around the problem and nd a clever argument that produces a logical contradiction. Which follows from the fact that every even number greater than $2$ is divisible by $2$, hence not prime. But proving contrapositive equivalent form is very easy, and you don't to do any choice.
Therefore y = 0, contradicting that it is positive. For our next example, consider the following proposition concerninganintegerx: Proposition If x2 ¡6 ¯5 iseven,thenx isodd.
6. The second approach works well for this problem.
The proof will use the following definitions. Example Questions. Let me show you another example where a contrapositive proof is so much easier to carry out.
We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. However, today we want try another approach that works well here and in other important cases where a contrapositive proof may not. Proposition: If is even then x is even. Prove that for x ∈ Z , if 5x + 9 is even, then x is odd.
Proof: Assume that x is odd then we have an integer k such that x = 2k+1. Then (x + y)(x y) = 1, so x y and x + y are divisors of 1. A contrapositive proof seems more reasonable: assume n is odd and show that n3 +5 is even. This is because it is easier to transforminformationabout xintoinformationabout7 ¯9 thantheother way around. Prove that if x is irrational, then x1=6 is also irrational.