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**AlephZero** The first part of the first question is just asking you to do some substitution of symbols for phrases. When you see "for all" or "for every," you substitute a $\displaystyle \forall$. Similarly, when you see "there exists" or "there is a," you substitute a $\displaystyle \exists$. So for the first one, we have $\displaystyle \forall x,$ $\displaystyle \exists y$ such that $\displaystyle 2xy=3.$ I'll let you do the second one.

The second part of the first question asks you if the statements are true or not. So (ii) asks whether it is true that there are two negative numbers which, multiplied together, result in another negative number. But of course we know that multiplying two negatives together makes a positive, and so this is false. I'll let you do (i) yourself.

The second question is about mathematical induction. This is a method used in math to prove certain statements about natural numbers. Suppose I want to prove that a certain statement is true. First, I show that it works for $\displaystyle n=1$. Then, I assume it is true for some arbitrary natural number $\displaystyle k$. I then show that this assumption implies that the statement must be true for $\displaystyle k+1.$

Try doing this for the example given. If you have problems, post again, show the work you've done, and I'll be happy to help you with any problems you may have.

Cheers