# Find the complement of the following expression and simplify: `x bar y + y bar z`Sorry i dont know how to do the not sign above the Z and y :/ .

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Expert Answers

lemjay | Certified Educator

`xbary+ybarz`

The complement of the above Boolean expression is:

> `bar(xbary+ybarz)`

To simplify, use De Morgan's Theorem which is `bar(A+B)=barA*barB` .

> `=bar(xbary)*bar(ybarz)`

Again, use De Morgan's Theorem to simplify further. The formula is `bar(AB)= barA+barB` .

> `= (barx+ bar(bary))*(bary + bar(barz))`

Note that `NOT(barA)=A` .

> `= (barx+y)*(bary+z)`

Then, distribute.

> `=barxbary + barxz + ybary + yz`

Apply the law of Boolean algebra which is `AbarA=0` .

> `= barxbary+barxz+yz`

**Hence, the complement of `xbary+ybarz` is `barxbary+barxz+yz` .**