20 min read

From Expression to Circuit Diagram

Break an expression into parts, draw the gates in the right order, and check with a truth table.

The Order of Operations

Boolean algebra has an order, like BODMAS in maths: 1 Brackets first. 2 NOT next. 3 AND next. 4 OR last. So in A + B \cdot C you do NOT C first, then B AND (NOT C), and finally A OR that result.

1 Write the inputs down the left of the page, one line each. 2 Draw a NOT gate for every letter that has a bar. 3 Draw the AND gates (and anything in brackets). 4 Draw the final OR (or whatever operation is done last) on the right. 5 Label each gate output and check it matches the expression.

X = 0

Try it:

Check that X = 1 whenever C = 1 and (A = 1 or B = 0).

The circuit for X = (A + B̄) \cdot C.

X = 0

Try it:

When A = 1 the output copies B; when A = 0 it copies C. (This circuit is called a multiplexer — it chooses between two inputs.)

The circuit for X = A\cdotB + A̅\cdotC. The dots show where wires split.

Practice

In X = A + B \cdot C̄, which operation is done first?

Drawing Method

1 Write the inputs down the left of the page, one line each. 2 Draw a NOT gate for every letter that has a bar. 3 Draw the AND gates (and anything in brackets). 4 Draw the final OR (or whatever operation is done last) on the right. 5 Label each gate output and check it matches the expression.

X = 0

Try it:

Check that X = 1 whenever C = 1 and (A = 1 or B = 0).

The circuit for X = (A + B̄) \cdot C.

X = 0

Try it:

When A = 1 the output copies B; when A = 0 it copies C. (This circuit is called a multiplexer — it chooses between two inputs.)

The circuit for X = A\cdotB + A̅\cdotC. The dots show where wires split.

Practice

In X = A + B \cdot C̄, which operation is done first?