Question

What is the equivalent of the following expression?

X′⋅Y′+X⋅Z′

In other words, what other boolean expression is the sadme as the one shown above.
X′⋅Y′+X⋅Z′=?

a. (X′+Z′)⋅(X+Y′)
b. X⋅(Y′+Z′)
c. (X+Z′)⋅(X′+Y′)
d. 1

Answer 1

The equivalent boolean expression for X′ ⋅ Y′ + X ⋅ Z′ is option c, which is (X+Z′) ⋅ (X′+Y′), as it simplifies to the original expression by applying the distributive property.

Step-by-step explanation:

The expression X′ ⋅ Y′ + X ⋅ Z′ can be analyzed using the properties of Boolean algebra to determine its equivalent expression.

Let's apply the distributive property of Boolean algebra, which is similar to the distributive property of ordinary arithmetic and vectors. According to the distributive property, A(B+C) = AB + AC.

Applying this to the given expression, we can see that none of the options directly applies the distributive property to result in the given expression. However, option c, (X+Z′) ⋅ (X′+Y′), can be expanded as XX′ + XY′ + XZ′ + Z′X′.

Remembering that any variable ANDed with its complement results in 0, the term XX′ equals 0, and Z′X′ simplifies to X′Z′ using the commutative property (A+B = B+A), we are left with XY′ + X′Z′, which is the original expression. Hence, option c is the correct equivalent expression.