2.1.9 exist and prove your answer. (c) Suppose that we change axiom (ii) to requ
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2.1.9
exist and prove your answer. (c) Suppose that we change axiom (ii) to require exactly four distinct points adjacent to any point but leave the other axioms unchanged. How many points must exist? Prove your answer. Generalize. *2.1.9. Consider the axiomatic system with the undefined terms apples, oranges, and likes and the following axioms. (i) There is an apple. (ii) No apple likes another apple. (iii) No orange likes another orange. (iv) Everything likes at least two other things. (v) Everything is an apple or an orange (a) Prove that nothing is both an apple and an orange. (b) Determine the minimum number of apples and oranges and prove your answer. (c) Add the following axiom to this system and redo part (b): If x likes y then y does not like x Consider the axiomatic system with the undefined terms point, circle, and on and the following axioms. 1.10.Explanation / Answer
a. Everything has to be an apple or an orange (axiom v). And everything has to like at least two other things (axiom iv). But if something is both an apple and an orange (negative of the conclusion), then either it would contradict axiom ii and axiom iii, hence conclusion is true (since negative of the conclusion contradicts)
b. There is an apple (axiom i). There has two be two other things for the apple to like (axiom iv). Neither of them can be apples because then the first Apple wouldn't like them (axiom ii). If they are both oranges, then the oranges can't like each other either (axiom iii), but they can like the apple. But each orange needs to like TWO other things (axiom iv again). So there must be a second apple for them to like. Now both apples like both oranges, and both oranges like both apples; all axioms are satisfied. So the minimum is 2 apples and 2 oranges.
c. Try to think of it in layers. Again, there is an apple in the original layer. We know there must be two oranges in the next layer for it to like. But each orange must now have TWO apples to like, and neither of them can be the original apple. So we add a new layer of 2 more apples for the oranges to like.
So now we have 1 apple, 2 oranges, and 2 more apples. But the new apples also need 2 things to like, and they can't like the previous oranges or the original apple. So we need 2 more oranges in a new layer.
So now we have 1 apple, 2 oranges, 2 apples, and 2 more oranges. The two new oranges need two things to like. They can't like back the previous layer of 2 apples and they can't like any preceding oranges. But they can like the original apple. So we need one more apple.
Now:
we have 1 apple, 2 oranges, 2 apples, 2 oranges, and 1 apple. The latest apple can't like any of the other apples, and it can't like-back the 2 preceding oranges. But it can like the previous 2 oranges. All our axioms are now satisfied. So the minimum is 4 apples and 4 oranges.
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