Boolean Identities
Suppose you are given two expressions, e.g., $(x + y)'$ and $x'y'$, and want to check if they are identical or not (i.e., to verify if $(x + y)' = x'y'$ is a valid identity or not.) You can do so by constructing the truth tables for each of the 2 expressions!
| $x$ | $y$ | $x + y$ | $\boldsymbol{(x + y)'}$ |
|---|---|---|---|
| $0$ | $0$ | $0$ | $1$ |
| $0$ | $1$ | $1$ | $0$ |
| $1$ | $0$ | $1$ | $0$ |
| $1$ | $1$ | $1$ | $0$ |
| $x$ | $y$ | $x'$ | $y'$ | $\boldsymbol{x'y'}$ |
|---|---|---|---|---|
| $0$ | $0$ | $1$ | $1$ | $1$ |
| $0$ | $1$ | $1$ | $0$ | $0$ |
| $1$ | $0$ | $0$ | $1$ | $0$ |
| $1$ | $1$ | $0$ | $0$ | $0$ |
Because the sequences of bits that we got under the $(x + y)'$ and $x'y'$ columns are exactly the same: $[1, 0, 0, 0]$, this means that the expressions are equivalent, so $(x + y)' = x'y'$ is a valid identity!
Bonus point: What is the name of this identity?
These notes by Miriam Briskman are licensed under CC BY-NC 4.0 and based on sources.