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Mathematical Logics - Essay Example

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This essay stresses that Mathematical logic can be defined as the applied mathematical formulas and techniques to solve logical problems. In the modern times, the mathematical logic includes model theory, set theory, recursion theory and proof theory…
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Mathematical Logics
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Introduction
Mathematical logics can be defined as the application mathematical formulas and techniques to solve logical problems. In the modern times, mathematical logic includes model theory, set theory, re-cursion theory and proof theory (Lavrov and Maksimova 170). Modal logics may have been used since time immemorial, especially by philosophers. In the modern times I believe that the most important Change in mathematical logic has been the development of many other kinds of logics which have supplemented the standard or classical one used in Mathematics. Mathematical logics help to bring formality in facts. It does not only promote proper reasoning but it also enhances a proper use of common sense (Lavrov and Maksimova 170).
From the set theory, our set in this question are the numbers between one and nine. The sub sets are: odd numbers, even numbers and individual numbers.
The sum of numbers available for Andy are Belle’s; 3+4+7=14 or Carol’s 4+6+8=18 or the sum on my cards=?Let the sum of Andy’s cards be A, Sum of Belles cards is B and sum of Carols cards be C and the sum of my cards is M. So M=B or M=C. This is because we are told that there are at least two card sums which are equal. The sum of Carols cards and Belle’s cards are not equal(B ≠ C).This means that the sum of my cards are either equal to Carol’s or Belle .This reasoning is derived from the fact that Andy admitted that there are two people with equal sum of cards.
The union of odd numbers is 1,3,5,7 and 9 .Also Belle admits that he can see all the five odd numbers, what does this imply? Firstly Belle cannot see 3 and 7 since he is the one possessing these subsets. Secondly it means that the odd numbers Belle sees are 1, 5 and 9.It is obvious that Andy possesses the card with 1 on it. Who has 9 and 5? Since there is no one else in the game, it is obvious that I possess cards with 9 and 5.
From above, the sum of my cards so far is 9+5=14.I still have another card to add to this which means the sum of my cards can never be 14.That is M>14.Therefore it is obvious that the sum of my cards is 18 since it is the only remaining (M=18).14+X=M=18.X=18-14=4.This means my third card is 4.Therefore my cards are 9, 5 and 4.
In conclusion from the above calculations and reasoning, it is evident beyond reasonable doubt that I have a 4, a 9 and a five. These answers were arrived after a rigorous process of extensive logical reasoning and application of a little bit of Boolean algebra. The fact that there were three sets of cards visible to Andy, that is mine, Carol and Belle enables me to come down to the situation that the actual cards available to Andy are two sets. Also the fact that there were only two available sums (14 and 18) makes it simpler for me to narrow on only two possibilities. Finally the sight of all odd numbers by Belle makes it easier to conclude that I got two odd numbers.
Reference
Lavrov Igor and Maksimova Larisa. Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms, Springer, 2003. Read More
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