[7 minutes] Using the various book and lecture Boolean identities, laws and expressions, minimize xy'+z+(x+y)z, to its lowest terms. When creating the function, use for not (ie. b), use xor for exclusive-or (ie. axorb), use xnor for exclusive-nor (ie.
b=(a'+b)(a+b')
Using DeMorgan's Law, we can simplify the expression xy' + z + (x + y)z as follows:
xy' + z + (x + y)z
= (xy' + z) + (x + y)z
= (xy' + z) + (xz + yz)
= xy' + z + xz + yz
Now, we can apply the distributive law to eliminate the parentheses:
xy' + z + xz + yz
= xy' + xz + yz + z
= x(y' + z) + (y + z)
Using the absorption law, we can further simplify:
x(y' + z) + (y + z)
= x + (y + z)
= x + y + z
So, the expression has been minimized to x + y + z.
To minimize the expression xy'+z+(x+y)z, let's simplify it step by step using the Boolean identities, laws, and expressions.
Step 1: Distribute the z term over the (x+y) term
(xy'+z) + z + (x+y)z
Step 2: Distribute the z over the (x+y) term using the distributive law
xy' + z + zx + zy
Step 3: Rearrange the terms to group similar ones together
(xy' + zx) + (zy + z) + z
Step 4: Apply the absorption law to xy' + zx
(xy' + zx) = (x + z)⋅(y' + z)
Step 5: Apply the absorption law to zy + z
(zy + z) = (z + y)
Step 6: Substitute the simplified terms back into the original expression
(x + z)⋅(y' + z) + (z + y) + z
Step 7: Apply the distributive law to (x + z)⋅(y' + z)
(xz + zz + xy' + y'z) + (z + y) + z
Step 8: Simplify using the idempotent law (z + zz = z)
(xz + z + xy' + y'z) + (z + y) + z
Step 9: Group similar terms together
xz + z + xy' + y'z + z + y + z
Step 10: Simplify the expression by combining like terms
xz + xy' + y'z + 3z + y
Therefore, the minimal expression for xy'+z+(x+y)z is xz + xy' + y'z + 3z + y.