Basic set theory and logic tutorial

  1. In the town of Seville lives a barber who shaves everyone who does not shave himself. Does the barber shave himself? In terms of set theory, the question is :  Let R be the set of all sets that are not members of themselves. Does R contain itself? (please do not Google else you will lose a golden chance to train your intellect. Think of the tutorial question as nurturing your intellect in math and logic…)
  2. Suppose X has 3 friends: A, B and C, who are respectively, good at cricket, music and mountaineering, but not at any other fields. What is the fallacy in the following? “The statement ‘One of X’s friends is a cricketer’ is true and so is the statement ‘One of X’s friends is a musician’. So their conjunction is true, that is, the statement ‘One of X’s friends is both a cricketer and a musician’ is true. But X has no such friends.
  3. Take an English-to-English dictionary (any other language will also do). Start with any word and note down any word occurring in its definition as given in the dictionary. Take this new word and note down any word appearing in its definition. Repeat the process with this new word until a vicious circle results. Prove that a vicious circle is unavoidable no matter which word one starts with . (Caution: the vicious circle may not always involve the original word) Note: So, now, do you see that the words “point” , “line” and “surface” are not defined, but taken as known “intuitively”? 
  4. On both the sides of a piece of paper it is written: “The sentence on the other side is false”. Are the two sentences so written statements? Why? What if on one side,”the sentence on the other side is true” is written and on the other side, “the sentence on the other side is false”? Note that in math: a statement can have only one truth value; it cannot contradict itself. 

Regards,

Nalin Pithwa.

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