Boolean expression $y.z+z$ is equal to which of the following?

Question

Accepted Answer

The correct option is (3):
The given Boolean expression is $y.z + z$. This expression can be elegantly simplified using one of the fundamental theorems of Boolean algebra, known as the **Absorption Law**. The Absorption Law states that for any two Boolean variables, $A + A.B = A$. In our expression, if we let $A$ be $z$ and $B$ be $y$, the expression is in the form $z + z.y$. Applying the Absorption Law directly simplifies $z + z.y$ to just $z$. Alternatively, a step-by-step simplification involves factoring $z$ out of the expression: $y.z + z = z.(y+1)$. According to the Dominance or Null Law, anything ORed with 1 is 1 (i.e., $y+1 = 1$), so the expression becomes $z.1$. Finally, the Identity Law states that anything ANDed with 1 is itself (i.e., $z.1 = z$), proving that the simplified and equivalent expression is $z$.

The incorrect option is (1):
The expression $y + y.z$ is actually a direct application of the Absorption Law itself, which simplifies this expression to $y$. Since the original expression, $y.z+z$, simplifies to $z$, and $y$ and $z$ are two independent Boolean variables which are not guaranteed to be equal for all input combinations, $y + y.z$ is not the correct equivalent. For example, if $y=0$ and $z=1$, the original expression is 1, but this option simplifies to $y=0$.

The incorrect option is (2):
The expression $y + z$ represents the logical OR operation between the two variables. Since the original expression $y.z+z$ simplifies strictly down to $z$, the two expressions, $y+z$ and $z$, are clearly not equivalent for all possible assignments of $y$ and $z$. If we test a case where $y$ is 1 and $z$ is 0, the original expression $y.z+z$ evaluates to $(1.0)+0 = 0+0 = 0$. However, the option $y+z$ evaluates to $1+0 = 1$. Because 0 is not equal to 1, the expressions are not logically equivalent.

The incorrect option is (4):
The expression $y.z$ represents the logical AND operation, which is only a subset of the conditions covered by $z$. We know the original expression $y.z+z$ simplifies to $z$. For $y.z$ to be equivalent to $z$, the variable $y$ would need to be permanently set to 1, as $1.z = z$. Consider the case where $y=0$ and $z=1$: the original expression $y.z+z$ evaluates to $(0.1)+1 = 1$, whereas $y.z$ evaluates to $0.1 = 0$. Since the results are different, $y.z$ is not the correct simplified form.