Write the dual of the following statement; ~ p ∧ ( q ∨ c ) .

The statement is ~ p ∧ ( q ∨ c ) Here, ∨ is the sign of disjunction, ∧ is the sign of conjunction and c is contradiction. To find the dual of the given statement, replace the conjunction sign ∧ with disjunction ∨ and disjunction sign ∨ with conjunction sign ∧ . And if there is any negation in the statement, do not change the negation of the statement. The duality of contradiction is tautology. So on replacing the connectives with each other, we get: ~ p ∨ ( q ∧ c )

Find the symbolic form of the following switching circuit. Construct its switching table and interpret your result.

Given switching circuit is Let, p : The switch S 1 is closed. q : The switch S 2 is closed. ~ p : The switch S 1 ' is closed or the switch S 1 is open. ~ q : The switch S 2 ' is closed or the switch S 2 is open. ∴ The symbolic form of the given circuit is: p ∨ q ∧ ~ p ∧ ~ q So, the switching table is as follows: p q ~ p ~ q p ∨ q p ∨ q ∧ ~ p ∧ ~ q T T F F T F T F F T T F F T T F T F F F T T F F In the above truth table, all the entries in the last column are ' F ' , ∴ The given circuit represents a contradiction. ∴ Irrespective of whether switches S 1 and S 2 are open or closed, the given circuit will always be open (i.e., off). Hence, no current will flow in the circuit irrespective of the position of the switches.

Express the following circuit in symbolic form:

Let p : the switch S 1 is closed. q : the switch S 2 is closed. r : the switch S 3 is closed. ~ p : the switch S 1 ' is closed or the switch S 1 is open. ~ q : the switch S 2 ' is closed or the switch S 2 is open. ~ r : the switch S 3 ' is closed or the switch S 3 is open. l : the lamp L is on. The symbolic form of the given circuit is: p ∨ ( q ∧ r ) = l l is generally dropped and it can be written as: p ∨ ( q ∧ r ) Input-Output table p q r q ∧ r p ∨ ( q ∧ r ) 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0