![]() ![]() ![]() What else do we know about r? It's area is c 2/4. So we know that the Pythagorean theorem holds and a 2+b 2=c 2. What do we know about r? It is a right triangle, with sides a and b, and hypotenuse c. So let's try to show that if r, then a=b! Remember: We cannot assume that a=b, because this is what we are trying to prove. In the rules of logic this is called the Chain Rule. This means that if we can show that if r, then a=b, then we have shown that if r, then i. What do we know about i? We know that a=b if and only if it is an isosceles triangle. Let's prove this statement by direct proof, since I believe that is the simplest way. In this case, by already assuming the triangle is isosceles and starting our proof from that, we would be proving (i->r) this is not equivalent to (r->i). One mistake that many people make in proofs is by assuming that the necessary part is true, then they go on to prove that when the necessary side is true, then the sufficient side is true. Contradiction is when you assume that the sufficient side is true and the necessary side is false, so in this case we'd assume r and ~i (r.~i), and go on with our proof to arrive at a contradiction or an absurdity in the proof. ![]() Contrapositive works as a proof for the conditional since (r -> i) is equivalent to (~i -> ~r) by the laws of logic. In this case we would be assuming (~i -> ~r), if ~i (i is false), then ~r (r is false). Direct proof is when you assume that the sufficient side is true, then prove that the necessary side is also true. In this case we would be assuming r is true, and arriving at the conclusion that i is also true. Contrapositive is when you assume the necessary part is false, and show that when the necessary side is false then the sufficient side is also false. There are 3 different ways to prove conditionals: by direct proof, contrapositive, and contradiction. Putting your statement into logic format, we have the conditional: (r -> i). Let i be "The right angled triangle t is an isosceles triangle." This is our necessary part of the conditional. Let r be "The right angled triangle t has side lengths a and b, hypotunese c, with area c 2/4." This is our sufficient part of the conditional. Since this is for Discrete Math, I'm assuming that what you are trying to do is prove this statement by the laws of logic. ![]()
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