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This morning, I completed a proof of the following proposition: there exist four distinct real numbers a,b,c,d such that exactly four of the numbers ab,ac,ad,bc,bd,cd are irrational. (IMO, this proposition is pretty cool). Note, the proof method to use here is proof by contradiction. Well, happy proof-writing AF members and anyone else.
Hint 1
In order to use contradiction, we must first take the negation of the conclusion that, there exist four distinct real numbers a,b,c,d such that exactly four of the numbers ab,ac,ad,bc,bd,cd are irrational (or equivalently, not rational). Thus, we can negate the statement via the following steps (we distribute the negation symbol through the statement from left to right by first tackling the quantifier "there is" and then tackling "the something that happens" in the statement. Note, the properties themselves stay the same):
i) ~[There exist four distinct real numbers a,b,c,d such that exactly four of the numbers ab,ac,ad,bc,bd,cd are not rational]
ii) For every four distinct real numbers a,b,c,d, ~[exactly four of the numbers ab,ac,ad,bc,bd,cd are not rational]
iii) For every four distinct real numbers a,b,c,d, exactly four of the numbers ab,ac,ad,bc,bd,cd are rational (taking the negation of a statement with the word "not" in it cancels out the "not"). Now, for simplicity, lets refer to this statement as A1, which is a new statement in the forward process: this process consists of statements that we create via facts and mathematical reasoning, which we can work forward from in order to establish the truth of the conclusion (note reasoning backwards from the conclusion via key questions, definitions, facts, etc., is referred to as the backward process.). Thus, since our proof is by contradiction, we must work forward from A1 in order to reach a contradiction (we do not have the luxury of working backward from anything).
Hint 2 (Analysis of Proof/Spolier)
A1: For every four distinct real numbers a,b,c,d, exactly four of the numbers ab,ac,ad,bc,bd,cd are rational.
Now, since, the keywords "for every" appear in A1, we can select any four distinct real numbers, say a=a',b=b',c=c',and d=d'. Thus, we have that
A2: a', b',c', and d' are four distinct real numbers.
Now, from A1, it follows that
A3: Exactly four of the numbers a'b', a'c', a'd', b'c', b'd', c'd' are rational.
Now, without loss of generality, assume that
A4: a'b', a'd', b'c', c'd' are rational.
Since a',b',c', and d' are distinct real numbers (via A2), then from A4, it follows that
A5: a',b',c', and d' are rational.
Now, the product of any two rational numbers is also rational, so it follows from A5 that
A6: a'b', a'c', a'd', b'c', b'd', c'd' must be rational.
But, A6 contradicts A3, which states that "exactly four of the numbers a'b', a'c', a'd', b'c', b'd', c'd' are rational."
Thus, we have reached a contradiction, which completes the proof (please note that the final product must be written in paragraph form in order to complete the proof-writing process).
P.S. I'm not a math guru/nerd/expert, so I apologize if my writing/reasoning is inelegant.