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I nee to determine whether these arguments are valid or not... How can i go abouts solving this question? I am having trouble finding a theoretical way to prove this... There are four of them, perhaps just a bit of help with one or two so i cant get the hang of it would be much appreciated! here they are

  • Everyone that can program in Java has a job. If Ryan can program in Python then he can also program in Java. Therefore, if Ryan can program in Python, then he has a job.

  • Everyone who loves waffles also loves muffins. Everyone that loves hashbrowns also loves waffles. A student loves waffles. Therefore, at least one student loves muffins.

  • Everyone that has a computer knows c++. Sue has a computer and John knows C++. Therefore, both Sue and John have a computer.

  • Every student has a tablet. John has a tablet. Tom does not have a tablet. Therefore, Tom is not a student or John is a student.*

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To prove "if $A$, then $B$," you assume $A$ and then try to prove $B$. So assume Ryan can program in Python. Then he can also program in Java, so he has a job.

To prove that at least one $x$ has property $P$, we look for an $x$ that might have property $P$. We know that a student loves waffles. That student must also love muffins, so at least one student loves muffins.

To prove that both Sue and John have a computer, you must prove that Sue has a computer and John has a computer. Sue has a computer, but we cannot prove that John has a computer. If John does not have a computer, but he knows C++, then all the hypotheses are still valid.

One method to prove "$A$ or $B$" is to either prove $A$ or to prove $B$. Just consider Tom and John separately and see if you can prove one of the two statements.

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The first statement: If you can program in Java, you have a job - this is the reformulation to make it easier to follow the train of logic. The argument is easily seen to be true then.

For the second: If you love waffles, you love muffins. If you love hashbrowns, you love waffles. A student loves waffles immediately implies the student loves muffins.

Basically you can change all statements into ones with logical operators (I've never taken logic before, but it's not too difficult to just write the words out).

Something to remember: converses are not generally true, contrapositives are ALWAYS true.

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