Translate each of the following into symbolic proof and provide the reason for each step: If I'm going to wear my umbrella, it's going to rain. Whenever it rains, I always wear a hat. Now, I didn't wear my hat. Therefore, it must not rain, so I do not carry my umbrella.You cannot be both happy and rich. Therefore, you are either not happy, or not rich. Now you do appear to be happy. Therefore, you must not be rich.If I were smart or good-looking, I would be happy and rich. But I am not rich. So it's true that either I'm not happy or I'm not rich. In other words, I am not both happy and rich. Therefore I am not smart or good-looking,. In other words I am not smart and neither am I good-looking. In particular, I am not smart.If interest rates fall, then the stock market will rise. If interest rates do not fall, then housing starts and consumer spending will fall. Now, consumer spending is not falling. So, it's true that housing starts are not falling or consumer spending is not falling; that is, it is false that housing starts and consumer spending are both falling. This means that interest rates are falling, so the stock market will rise.

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10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Translate each of the following into symbolic proof and provide the reason for each step:

If I'm going to wear my umbrella, it's going to rain. Whenever it rains, I always wear a hat. Now, I didn't wear my hat. Therefore, it must not rain, so I do not carry my umbrella.
You cannot be both happy and rich. Therefore, you are either not happy, or not rich. Now you do appear to be happy. Therefore, you must not be rich.
If I were smart or good-looking, I would be happy and rich. But I am not rich. So it's true that either I'm not happy or I'm not rich. In other words, I am not both happy and rich. Therefore I am not smart or good-looking,. In other words I am not smart and neither am I good-looking. In particular, I am not smart.
If interest rates fall, then the stock market will rise. If interest rates do not fall, then housing starts and consumer spending will fall. Now, consumer spending is not falling. So, it's true that housing starts are not falling or consumer spending is not falling; that is, it is false that housing starts and consumer spending are both falling. This means that interest rates are falling, so the stock market will rise.

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