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EE1001 Foundations of Digital Techniques

Assignment 2

Full mark: 16 points

1.   (2 points) Use a truth table to determine whether p ^ (q v r) and (p ^ q) v (p ^ r) are logically equivalent.

2.   (2 points) Use De Morgan’s laws to write the negations for the following statements:

a)    The train is late or my watch is lost.

b)   一2 < x < 7.

3.    (4 marks) Use Theorem 2.1 in the lecture notes to verify the logical equivalences

below. Supply a reason for each step.

a)   ~(pv ~q) v (~pΛ ~q) = ~p

b)  (pΛ (~(~pv q))) v (pΛ q) =p

4.  Without using truth tables, show the following:

a)  (1 point)  pv (pΛ q) =p

b)  (3 points) ((p一q) Λ p) 一q is a tautology

5.   (4 points) We visit an island in which every inhabitant is one of two types, knights

or knaves. Knights make only true statements while knaves make only false statements. We consider two inhabitants A and B. A is asked to say something about himself and B.

a)  Suppose A says,“Both of us are knaves.”What can be determined about the types of A and B? Use a truth table to explain your answer.

b)  What if, instead, A says,“Band I are of the same type,i.e., either both of us are knights or both of us are knaves.”What can be determined about the   types of A and B? Use a truth table to explain your answer.