### Getting Started

Combinational Logic uses a combination of basic logic gates AND, OR and NOT to create complex functions. For each output, the design procedure is:

- Derive the truth table.
- Simplify the boolean expression using Karnaugh Map (K map).
- Draw a logic diagram that represents the simplified Boolean expression. Verify by analysing or simulating the circuit.

Your objective is increase your score in the 3 areas: Truth Table, K Map, and Logic Gates while designing different digital circuits

You need to sign in with your Google account to save your progress.

hide/show Getting Started Information.

hide/show Topical Help.

Report a Problem.

## Specification

## Black Box Continue

A black box shows the relationship between its inputs and outputs without knowing its implementation.

Click on the inputs and observe the output(s) to understand the specification.

- When the input, output or wire is grey, its value is 0 (false).
- When the input, output or wire is green, its value is 1 (true).

## Truth Table Check Skip

Your task is replace all the **?** in the output column by clicking on the cell. When you are done click **Check**.

Hints

- When you click on the inputs of the blackbox, the corresponding row is highlighted in the truth table.
- On the truth table, note the input variables and their values.
- Recall that when the input, output in the blackbox is grey, its value is 0. When it is green, its value is 1.
- Use the output color to determine if the output cell should be 0 (grey) or 1 (green).
- If the blackbox has more than one output, we will solve one output at a time.

The Truth Table shows the values of the circuit output for all input values. The number of rows is 2^{n} where n is the number of inputs. So for 2 inputs, there are 4 rows and for 3 inputs there are 8 rows and so on.

You can also fill up the truth table using the specification.

Your task is to select the minterm(s) that represent **?** in the Sum of Product. When you are done, click **Check**

Hints

- Hover or tap on
**?**to find the respective output cell. - Study the other minterms to understand how they are formed.

There are 2 steps to obtain the Sum of Products from its truth table.

- A Minterm is a product • (AND) term containing all input variables. A variable X appears in complemented form X if it is a 0 in the row of the truth-table, and as a true form X if it appears as a 1 in the row.
- The Sum of Products is obtained by taking the sum + (OR) of the minterm of the rows where a 1 appears in the output.

## K Map Check Skip

Your task is to replace all the **?** in the K Map by clicking on that cell. When you are done, click **Check**

Hints

- When you hover or tap on an output cell in the truth table, the corresponding cell in the K Map is highlighted.
- Understand the relationship by studying the variables in the headers of both the truth table and the K map.

The Karnaugh map (K map) provides a simple and straight-forward method of minimising boolean expressions.

A K map is a two-dimensional truth-table. Note that the squares are numbered so that any two adjacent cells differ by a change in only one variable.

Your task is to memorise and group adjacent cells containing 1's.

- When you hover or tap on the product terms in the Boolean expression, you will see groups of adjacent 1's. Memorise all of them and click
**Hide**. - Now click on the cells containing 1's to group them as memorised earlier. When you have selected a group, click
**Group**. - Repeat the previous step until the button changes to
**Check**.

Rules for Grouping together adjacent cells containing 1's

- Groups must contain 1, 2, 4, 8, 16 (2
^{n}) cells. - Groups must contain 1's only.
- Groups may be horizontal or vertical, but not diagonal.
- Groups should be as large as possible.
- Each cell containing a 1 must be in at least one group.
- Groups may overlap.
- Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.
- There should be as few groups as possible.

Your task is to select the product term(s) that represent **?** in the Boolean Expression. When you are done, click **Check**

Hint

- Hover or tap on
**?**to find the respective group. - Study the other product terms to understand how they are formed.

Obtaining Product Terms

- If X is a variable that has value 0 in all of the squares in the grouping, then the complemented form X is in the product term.
- If X is a variable that has value 1 in all of the squares in the grouping, then the true form X is in the product term.
- If X is a variable that has value 0 for some squares in the grouping and value 1 for others, then it is not in the product term.

## Logic Gates Check Skip

Your task is to connect up the logic gates to realise the Boolean Expression obtained from the K Map. When you are done, click **Check**

Hint

- Hover or tap on the gate to find the boolean expression.
- Study the connected gate to understand its function.
- To connect, click from the output of an element and drag to the input of the element that needs to be connected.
- If you are using a mobile device, select zoom/pan if you need to zoom or pan the logic diagram and select wire if you want connect the circuit.

Connecting the logic gates

- An AND gate performs the same function as product • operator in boolean algebra. It has only 1 output but can have 2 or more inputs.
- An OR gate performs the same function as sum + operator in boolean algebra. It has only 1 output but can have 2 or more inputs.
- A NOT gate inverts its input. It has only 1 output and only 1 input. If the input is X, then its output is X

## Design Complete Continue

### Truth Table

There are three ways to describe a Boolean function: logic circuit, truth table, and Boolean expression.

To convert a truth table to its circuit

- Convert the truth table to a Boolean expression using the sum of products technique.
- Minimise the Boolean expression using Karnaugh map (or Boolean algebra).
- Convert the minimised Boolean expression into a circuit.