Bebras 2022 - Level 5 Question 13

A truth table produces a white or black circle as output based on one or more input values. In the left-hand figure below, a table describes the input values of $x,y,z$ and the corresponding output.



A simpler representation is possible using the table on the right. For example, if input $x$ is black, the output must be black. If $x$ is white and inputs $y-z$ are white–black, the output is black.

The following table describes inputs $(x,y,z,t)$ and the output.



Which simplified table below agrees with the given table?

Editable data: 0 = white circle, 1 = black circle.

Example truth table:

| x | y | z | Output |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |

Simplified example:

| yz / x | 0 | 1 |
|---|---|---|
| 00 | 0 | 1 |
| 01 | 1 | 1 |
| 10 | 0 | 1 |
| 11 | 0 | 1 |

Target truth table:

| x | y | z | t | Output |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 0 |