As an award winning ph.d. Student in mathematics , I have written thousands of proofs. Usually, they are in analysis or topology or algebra.... Honestly, I kind of resent being asked to prove something this elementary / obvious . The point being disputed by some is that votes cast for a third party candidate could if they were cast for President Obama make the difference between the President winning and the Republican winning.
I am not speaking to the probability of the event and this outcome . I am simply saying that it is conceivable that the Republican candidate for President could win a state that puts him over 270 electoral votes by fewer votes than were cast for the third party candidates in the state. Moreover, it is also possible that the republican could win a state that puts him over 270 by fewer votes than were not cast by registered voters (ie not voting) in that state .
Now, this seems so transparently obvious that one might conclude that anyone who seriously disputes the thesis simply cannot add. Nevertheless, for the benefit of the less cognitively fortunate among us, we proceed.
Suppose that not all of the votes in the 2012 general election for president have been counted and neither the republican candidate for president nor President Obama has yet won enough states to put him over the 270 electoral. It's threshold. Suppose that the one contested state which has yet to be awarded has enough electoral votes that whoever wins it will be put over the 270 electoral vote mark by winning that particular state. The definitions which follow all pertain to the votes within that one contested state.
Let a be an element of the natural numbers such that a = the number of votes cast for the Republican candidate for President. Let b be an element of the natural numbers such that b = the number of votes cast for President Obama. Let c be an element of the natural numbers such that c = the number of votes cast for president in 2012 that were not cast for either President Obama or cast for the republican candidate for president. Let d = the number of votes failed to be cast by registered voters in that state.
what must be shown is that there exists a, b,c, and d as defined above such that a > b but a < b + c + d . One example with natural numbers a, b, c, and d is sufficient to prove a there exists statement. So, we pick a= 10 b=6 c = 3 d = 4. Clearly, the republican would win since 10 > 6. However, had all the third party votes been cast instead for President Obama and had all the registered voters who did not vote decided to cast their vote for President Obama then President Obama would have win that state. Since if President Obama had won that state he would have won that state and thus the election, we may conclude that the voters who chose to not vote or vote for a third party candidate in part caused the Republican to win and President Obama to lose reelection . QED.
8:58 PM PT: Since some still are having issues: minus negative one is equivalent to positive one. If the negative votes are the Obama votes and the positive votes are the republican votes, then taking one vote from President Obama is, in effect, the same as adding a republican vote.
8:58 PM PT: Since some still are having issues: minus negative one is equivalent to positive one. If the negative votes are the Obama votes and the positive votes are the republican votes, then taking one vote from President Obama is, in effect, the same as adding a republican vote.