Question

Which of the following is correct?
(a) \[\left( p\vee q \right)\equiv \left( p\wedge q \right)\]
(b) \[\left( p\to q \right)\equiv \left( q\to p \right)\]
(c) \[\left( p\to q \right)\equiv \left( p\wedge q \right)\]
(d) None of these

Answer 1

Hint: Draw a truth table for each expression provided in the options using the properties of Boolean Algebra. Check if the L.H.S. of the expression for particular values of p and q is equal to the R.H.S. of the same expression for the same particular values of p and q or not. If they are equal then the expression will be equivalent otherwise not.

Complete step by step answer:
Here, we have been provided with some Boolean expressions in the given options and we have to determine the correct option in which the two sides of the expression are equivalent. So, let us draw truth tables for each option and check them – by – one.
(a) \[\left( p\vee q \right)\equiv \left( p\wedge q \right)\]
Drawing the truth table for the above expression, we get,

p	q	\[p\vee q\]	\[p\wedge q\]
T	T	T	T
T	F	T	F
F	T	T	F
F	F	F	F

Clearly, we can see that the truth values for the expressions \[\left( p\vee q \right)\] and \[\left( p\wedge q \right)\] are not the same. So, we have L.H.S \[\ne \] R.H.S. therefore, the two expressions are not equivalent.
(b) \[\left( p\to q \right)\equiv \left( q\to p \right)\]
Drawing the truth table for the above expression, we get,

p	q	\[p\to q\]	\[q\to p\]
T	T	T	T
T	F	F	T
F	T	T	F
F	F	T	T

Clearly, we can see that the truth values of the expressions \[\left( p\to q \right)\] and \[\left( q\to p \right)\] are not the same for particular values of p and q. so, we have L.H.S \[\ne \] R.H.S. therefore, the two expressions are not equivalent.
(c) \[\left( p\to q \right)\equiv \left( p\wedge q \right)\]
Drawing the truth table for the above expression, we get,

p	q	\[p\wedge q\]	\[p\to q\]
T	T	T	T
T	F	F	F
F	T	F	T
F	F	F	T

Clearly, we can see that the truth tables for the expressions \[\left( p\to q \right)\] and \[\left( p\wedge q \right)\] are not the same. So, we have L.H.S \[\ne \] R.H.S. Therefore, the two expressions are not equivalent.
On observing all the truth tables for the expressions provided in each option we can conclude that none of the three options have equivalent expressions on both sides.
Hence, option (d) is the correct answer.
Note:
 One may note that the truth table for the expression in option (b) can confuse us. Here, we have three T’s for both the expressions \[p\to q\] and \[q\to p\] and only one F. But that does not mean both the expressions are equivalent because the truth values are not the same for corresponding values of p and q. You must remember the truth tables for basic expressions like: - \[p\vee q\], \[p\wedge q\], \[p\to q\] and \[p\leftrightarrow q\] as they are used everywhere.