Logic Gatekeeper

Three notations, one idea

Boolean logic looks like three separate topics and is really one. A question might show you a circuit made of gate symbols, or a line of algebra like Q = A AND B, or a table of ones and zeros. Those are three ways of writing exactly the same thing, and the reason the topic exists is that computers are built out of it: every decision a processor makes comes down to combinations of these operations. Here is the useful part. Learning what AND does takes about a minute. What the exam actually tests is your ability to move between the three notations, so that when you are given a diagram you can produce the expression, and when you are given an expression you can complete the table. That is the skill this module builds. It is also a small topic, more so on your specification than on some others, and it can be finished properly rather than half-known.

The three OCR examines

Your specification examines three logic operators. Not four, not six. Learn these three properly and the topic is covered.

Notation to notation

Only three gates

Revising from a textbook written for a different exam board, you meet a gate called XOR, whose output is 1 when the inputs are different. What should you do?

• Note that it exists and move on, because your specification examines AND, OR and NOT only
• Learn its truth table thoroughly, since it may come up
• Use it in your answers wherever it makes an expression shorter
• Assume the textbook is wrong, since XOR is not a real gate

Reading a diagram in words

You cannot draw a circuit in an answer booklet without knowing how to read one first. Work left to right, and name each signal as it appears. Here are the two directions you will be asked to travel.

Which two are correct

Select the TWO statements that are true.

• For Q = NOT (A OR B), the output is 1 only when A and B are both 0
• For Q = (A OR B) AND NOT A, the output is 1 only when A is 0 and B is 1
• An OR gate gives 0 when both of its inputs are 1
• A circuit with three inputs needs six rows in its truth table

Column by column

Here is the method your specification is pointing at when it says to work multi-gate tables a column at a time, and it is the difference between getting these questions right and guessing at them. Do not try to evaluate a whole expression in your head for each row. Instead, give every bracket its own column. Take Q = (A AND B) OR (NOT C). That expression has two brackets, so your table gets two middle columns, one for A AND B and one for NOT C, before the Q column at the end. Now fill the table downwards rather than across: complete the whole A AND B column first, thinking about nothing but AND. Then complete the whole NOT C column, thinking about nothing but NOT. Only then do you fill Q, and by that point you are just applying OR to two columns already sitting in front of you. Three easy passes instead of eight hard ones, and if you make a mistake it is confined to one column where you can find it. Set the rows out by counting up in binary so that you cannot miss a combination: with three inputs there are eight rows, and an answer with seven is wrong before you start.

Fill the combined circuit

Complete the truth table for Q = (A AND B) OR (NOT C). Fill the two middle columns first, then the output. All eight input combinations are listed for you.

Name the middle column

To build a truth table for a circuit with more than one gate, first count the inputs: two inputs give _____ rows and three give _____. List the combinations by counting up in _____. Then add one column for each _____ in the expression, and complete those columns before the output column. Filling the table _____ rather than across means each pass involves only one operator, and a mistake stays where you can find it.

The expression that is not equivalent

Four statements about a circuit where A and B go into an OR gate whose output is then reversed. Select the ONE that is incorrect.

• The circuit can be written as Q = NOT (A OR B).
• Its output is 1 only when both A and B are 0.
• It is the same as Q = (NOT A) OR (NOT B).
• Its truth table has four rows, because there are two inputs.

Say what the circuit does

A circuit takes three inputs. A and B meet at an AND gate. C passes through a NOT gate. Those two results meet at an OR gate to give Q. Assemble the expression.

Turn the diagram into a table

An exam question shows a circuit with inputs A, B and C and asks you to complete its truth table. Take the decisions in order.

• Before anything else, how many rows will your table need, and why?
• How should you set out the input combinations?
• The expression is Q = (A AND B) OR (NOT C). What columns should the table have?
• In what order should you fill the cells?

Explain the three notations

A friend can recite what AND, OR and NOT do but freezes when given a circuit with two gates in it. Explain how to get from a diagram to a completed truth table.

• Name the three logic operators your specification examines
• State what each one outputs, and say which has only one input
• Explain what a logic diagram, a Boolean expression and a truth table each are
• Explain how to read a diagram from left to right to build an expression
• Explain how many rows a truth table needs, and why, for two inputs and for three
• Explain why the input combinations are listed by counting up in binary
• Describe the column-by-column method for a multi-gate circuit, and say why it beats working row by row
• Give an example expression with two brackets and say which columns its table would have