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### High School Mathematics - 214.4 Tautologies

A statement that is always true for all logical possibilities is called a tautology.
A statement that is always false is called a contradiction.
Thus the truth table of a tautology will have T and only T in the last column while a contradiction will have F and only F in the last column. Hence to determine whether a given statement is tautology or not, we construct the truth table and see whether the last column contains all T's or not.
All tautologies are true by virtue of their logical structure, and their tuth is independent of anything to which the sentences might refer. In other words, a tautology is logically true, and its truth has nothing to do with whatever the sentence may be talking about.

Example: Prove that the sentence ' It is wet or it is not wet' is a tautology.
Solution: The sentence may be symbolised by p v ~p where p symbolises the sentence 'It is wet '. The truth table is as shown below.
 p ~q p v ~q T F T F T T

 Q 1: Use truth table to establish which of the following statements is tautology or contradiction. p v ~(p ^ q)ContradictionTautology Q 2: Answer: Q 3: Use truth table to establish which of the following statements is tautology or contradiction. (p v q) v (~p)ContradictionTautology Q 4: Use truth table to establish which of the following statements is tautology or contradiction. ~[p v (~p)]ContradictionTautology Q 5: Use truth table to establish which of the following statements is tautology or contradiction. [(~p) v q] v [p ^ (~q)]TautologyContradiction Q 6: Use truth table to establish which of the following statements is tautology or contradiction. ~[p ^ (~p)]ContradictionTautology Q 7: Use truth table to establish which of the following statements is tautology or contradiction. (p ^ q) ^ ~(p v q)ContradictionTautology Q 8: Show by means of a truth table that the statement p v ~p is a tautology and the statement p ^ ~p is a contradiction.Answer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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