Question

Draw the \({\bf{3}}\)-cube \({{\bf{Q}}_{\bf{3}}}\) and label each vertex with the minterm in the Boolean variables \({\bf{x, y}}\), and \({\bf{z}}\) associated with the bit string represented by this vertex. For each literal in these variables indicate the \({\bf{2}}\)-cube \({{\bf{Q}}_{\bf{2}}}\) that is a subgraph of \({{\bf{Q}}_{\bf{3}}}\) and represents this literal.

Short answer

Top of the \({\bf{2}}\) cube: \({\bf{x}}\) (other variable changes)

Bottom of the \({\bf{2}}\) cube: \({\bf{\bar x( - do - )}}\)

Left side of \({\bf{2}}\) cube: \({\bf{\bar z}}\)

Right side of \({\bf{2}}\) cube: \({\bf{z}}\)

Rear side of \({\bf{2}}\) cube: \({\bf{y}}\)

Front side of \({\bf{2}}\) cube: \({\bf{\bar y}}\)

Step-by-step solution

Step 1:Definition

A Cube is a solid three-dimensional figure, which has six square faces, eight vertices and twelve edges. It is also said to be a regular hexahedron.

Variables of the cube

The required diagram for the \({\bf{3}}\)-cube is:

Top of the \({\bf{2}}\) cube: \({\bf{x}}\) (other variable changes)

Bottom of the \({\bf{2}}\) cube: \({\bf{\bar x( - do - )}}\)

Left side of \({\bf{2}}\) cube: \({\bf{\bar z}}\)

Right side of \({\bf{2}}\) cube: \({\bf{z}}\)

Rear side of \({\bf{2}}\) cube: \({\bf{y}}\)

Front side of \({\bf{2}}\) cube: \({\bf{\bar y}}\)