Question

In a Boolean algebra B, for all x, y in B, \[x \wedge (x \vee y)\] is equal to
A. y
B. x
C. 1
D. 0

Answer 1

Hint: We use laws of Boolean algebra to solve the given equation. Firstly use distributive law to break the equation into two terms. Use idempotent law to make the equation simpler. In the end use the absorption law to solve the final term.
* Conjunction: The logical operation of joining two statements with the operator ‘and’ in between is called conjunction. It is denoted by the symbol (\[ \wedge \]).
* Disjunction: The logical operation of joining two statements with the operator ‘or’ in between them is called a disjunction. It is denoted by the symbol (\[ \vee \]).

Complete step-by-step answer:
We are given the equation\[x \wedge (x \vee y)\].............… (1)
We can break the equation into two parts using distribution law of Boolean algebra
Since we know distribution law states that\[A.(B + C) = (A.B) + (A.C)\], so the equation becomes
\[ \Rightarrow x \wedge \left( {x \vee y} \right) = \left( {x \wedge x} \right) \vee \left( {x \wedge y} \right)\].........… (2)
Since the first bracket contains both the same elements with an operation between them, we can apply idempotent law on it.
We know idempotent law states that\[A.A = A\]
\[ \Rightarrow x \wedge x = x\]
 Substitute the value of\[x \wedge x = x\]in equation (2)
\[ \Rightarrow x \wedge \left( {x \vee y} \right) = x \vee \left( {x \wedge y} \right)\] ………...… (3)
We can clearly see that the RHS of the equation is at a place where we can apply absorption law.
Since we know absorption law states that\[A.(A + B) = A\]
\[ \Rightarrow x \vee \left( {x \wedge y} \right) = x\]...........… (4)
Substitute the value from equation (4) in equation (3)
\[ \Rightarrow x \wedge \left( {x \vee y} \right) = x\]
\[\therefore \]The value of \[x \wedge \left( {x \vee y} \right)\] is equal to x

\[\therefore \]Correct answer is option B.

Note: Boolean algebra: Boolean algebra is a branch of algebra that deals with mathematical logic. Values of the variables are given by truth tables.
* Laws of Boolean algebra are:
1) Distributive Law: Distributive law states that\[A.(B + C) = (A.B) + (A.C)\] and \[A.(B + C) = (A + B).(A + C)\]
2) Idempotent Law: Idempotent law states that \[A + A = A\] and \[A.A = A\]
3) Absorption Law: Absorption law states that \[A.(A + B) = A\] and \[A + AB = A\]