The Boolean expression XY + ( X′ + Y′) Z is equivalent to

Boolean Expression Simplification

To determine the equivalent Boolean expression, we will simplify the given expression step-by-step using fundamental Boolean algebra laws. Then, we will check which of the provided options simplifies to the same result.

Given Boolean Expression

The given Boolean expression is: $XY+(X^{\prime }+Y^{\prime })Z$

Step-by-Step Simplification of the Expression

1. Applying De Morgan's Theorem

The term $(X^{\prime }+Y^{\prime })$ can be simplified using De Morgan's Theorem, which states that $(A^{\prime }+B^{\prime })=(AB{)}^{\prime }$. Applying this to our expression:

$(X^{\prime }+Y^{\prime })=(XY{)}^{\prime }$

Substitute this back into the original expression:

$XY+(XY{)}^{\prime }Z$

2. Applying the Absorption Law/Identity

Now, we use a key Boolean algebra identity: $A+A^{\prime }B=A+B$. In our current expression, let A = XY and B = Z. Applying this identity:

$XY+(XY{)}^{\prime }Z=XY+Z$

So, the given Boolean expression $XY+(X^{\prime }+Y^{\prime })Z$ simplifies to $XY+Z$.

Simplifying the Correct Option for Equivalence

The correct answer option provided is $(X+Z)(Y+Z)$. Let's expand and simplify this expression to see if it matches our result of $XY+Z$.

1. Applying the Distributive Law

Expand the product $(X+Z)(Y+Z)$ using the Distributive Law $A(B+C)=AB+AC$:

$(X+Z)(Y+Z)=X(Y+Z)+Z(Y+Z)$

$=XY+XZ+ZY+ZZ$

2. Applying the Idempotent Law

According to the Idempotent Law, $A\cdot A=A$. Therefore, $ZZ=Z$.

Substitute this back into the expression:

$=XY+XZ+YZ+Z$

3. Applying Absorption Property (repeatedly)

Now we simplify $XY+XZ+YZ+Z$. We can use the absorption property $A+AB=A$ or $A+B=A+B(1)=A+B(C+1)$, and $A+1=1$. A simpler way to look at this is that if a variable (like $Z$) is present as a standalone term in a sum, any term that is a product involving that variable will be absorbed by the standalone variable.

- Consider $XZ+Z$: This simplifies to $Z(X+1)$. Since $X+1=1$, then $Z(1)=Z$.
- Similarly, $YZ+Z$: This simplifies to $Z(Y+1)$. Since $Y+1=1$, then $Z(1)=Z$.

Applying this to our expression:

$XY+XZ+YZ+Z$

Group terms with $Z$:

$=XY+(XZ+Z)+YZ$

$=XY+Z+YZ$

Again, group terms with $Z$:

$=XY+(Z+YZ)$

$=XY+Z$

Conclusion on Equivalence

We found that the original Boolean expression $XY+(X^{\prime }+Y^{\prime })Z$ simplifies to $XY+Z$. We also found that the option $(X+Z)(Y+Z)$ simplifies to $XY+Z$.

Since both expressions simplify to the same form, $XY+Z$, they are equivalent.

Summary of Boolean Algebra Laws Used