Prove the absorption law x + xy = x using the other laws in Table 5.
The Correct Answer and Explanation is :
To prove the absorption law ( x + xy = x ) using Boolean algebra, we can apply several fundamental laws. Here’s a step-by-step derivation:
- Distributive Law: actor out ( x ) from the left-hand side: [
x + xy = x(1 + y)
]
his step utilizes the distributive property, which states that ( a + ab = a(1 + b) ). - Domination Law: implify ( 1 + y ) to 1: [
x(1 + y) = x \times 1
]
ccording to the domination law, ( 1 + y = 1 ) for any Boolean variable ( y ). - Identity Law: implify ( x \times 1 ) to ( x ): [
x \times 1 = x
]
he identity law states that ( a \times 1 = a ) for any Boolean variable ( a ).
hus, we’ve shown that ( x + xy = x ), confirming the absorption law.
Explanation of the Laws Used:
- Distributive Law: his law allows us to factor expressions in Boolean algebra. It states that ( a + ab = a(1 + b) ), which is analogous to factoring in regular algebra.
- Domination Law: his law simplifies expressions involving the OR operation. It states that ( 1 + y = 1 ) for any Boolean variable ( y ). This is because, in Boolean logic, the OR operation with 1 always results in 1.
- Identity Law: his law simplifies expressions involving the AND operation. It states that ( a \times 1 = a ) for any Boolean variable ( a ). This is because, in Boolean logic, the AND operation with 1 leaves the other operand unchanged.
y applying these laws, we can simplify complex Boolean expressions and prove various identities, such as the absorption law.