GCSE · Computer Science · Edexcel · Spec 1CP2

OR in truth tables

Three inputs, eight rows. How many of them output 0? Guess now, then see why you'll never need to work an OR table out the slow way again.

Build it row by row

Hunt for the zero

Three inputs, eight rows. The A OR B column is your working: it's the two-input rule, done first. Then OR it with C.

Keep your guess about the zeros in mind as you step through.

ORA OR B OR Cgate

A	B	C	A OR B	A OR B OR C
0	0	0	0	0
0	0	1	0	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	1	1

A=0, B=0, C=0, A OR B=0 → A OR B OR C = 0

Output

Step 1: Row 000. Start with the two-input rule: A OR B is 1 if A is 1, B is 1, or both are. Here A and B are both 0, so A OR B is 0. Now OR that with C. C is 0 too, so the output is 0. Every input is 0. Hold that thought.

1 / 7

Step through with Next, or press Play and watch where the lone 0 sits.

Step 1 of 7: Row 000. Start with the two-input rule: A OR B is 1 if A is 1, B is 1, or both are. Here A and B are both 0, so A OR B is 0. Now OR that with C. C is 0 too, so the output is 0. Every input is 0. Hold that thought..

Either, or both?

Where does the 11 row go?

Each chip is one row of the A OR B table. For each row, tick the circle (or circles) it belongs in. Is A 1 in this row? Is B?

A A is 1
B B is 1

A = 0, B = 0
A = 0, B = 1
A = 1, B = 0
A = 1, B = 1

Predict, then check

Brackets are evaluated first. Commit to an answer before you build anything.

Here's A OR (A AND B). For AB = 00, 01, 10, 11, what will its output column be?

Your turn to drive

Build A OR (B AND C)

Build the truth table for A OR (B AND C). Choose each missing step.

Three inputs, so the table needs eight rows.
missing step

Which line is step 2?

List ABC like the two-input table: 000, 001, 010, 011.List ABC as binary numbers in order: 000, 001, 010, 011, 100, 101, 110, 111.List all-off, each input on by itself, then all-on: 000, 100, 010, 001, 111.

Spot the slip

Where did this table go wrong?

A student builds the truth table for NOT A OR B. Find the line where it goes wrong.

A student's working — which line goes wrong?

Inputs in binary order: AB = 00, 01, 10, 11.
Work out A OR B first: 0, 1, 1, 1.
Apply NOT to that column: 1, 0, 0, 0.
So NOT A OR B = 1, 0, 0, 0.

Put it to work

Does the circuit do what was intended?

In a circuit, inputs C and I go into an OR gate, and the OR gate's output goes into a NOT gate. The circuit was meant to output 1 only when C is 1 and I is 0. Use a truth table to decide whether the circuit does what was intended, and name a row that shows it. [4 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

Labels the gate outputs: C OR I from the OR gate, then NOT (C OR I) as the final output. (1 mark)
C OR I column for CI = 00, 01, 10, 11: 0, 1, 1, 1. (1 mark)
NOT (C OR I) column: 1, 0, 0, 0, set beside the intended column 0, 0, 1, 0. (1 mark)
Concludes the circuit does not do what was intended, naming a row that differs: CI = 10 (intended 1, circuit gives 0) or CI = 00 (intended 0, circuit gives 1). (1 mark)

Model answer

Label the OR gate's output C OR I, and the NOT gate's output NOT (C OR I). For CI = 00, 01, 10, 11, C OR I is 0, 1, 1, 1, so NOT (C OR I) is 1, 0, 0, 0. The intended output is 0, 0, 1, 0. The columns don't match, so the circuit does not do what was intended. Row CI = 10 shows it: the intended output is 1, but the circuit gives 0. (Row CI = 00 fails too: the circuit gives 1 where 0 was intended.)

A truth table turns 'does this circuit work?' into a row-by-row comparison. One row that disagrees is enough to show the circuit is wrong.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

The OR rule: the output is 1 if either one or both inputs is 1, and 0 only when every input is 0.
A truth table shows the output for every possible combination of inputs. 1 means true and 0 means false.
Two inputs need four rows and three inputs need eight. Write the rows as binary numbers in order so that none are missed.
Order of precedence: brackets are evaluated first, and NOT is evaluated before AND and OR.

The big picture

OR outputs 1 if either one or both of its inputs is 1, and 0 only when every input is 0. To build a truth table, list every combination of inputs in binary order, give each part of the expression its own column, and evaluate brackets first and NOT before OR. A truth table is also how you check whether a circuit does what was intended.

Key points

1A OR B reads 0, 1, 1, 1 for AB = 00, 01, 10, 11. The row where both are 1 is still a 1.

2A OR B OR C has one 0 row (000) and seven 1s. With only ORs, the order you combine them in doesn't matter.

3NOT A OR B means (NOT A) OR B and gives 1, 1, 0, 1. NOT (A OR B) inverts the whole OR and gives 1, 0, 0, 0.

4A OR (A AND B) gives 0, 0, 1, 1, the same as A, whatever B is.

5For a circuit, label each gate's output with its Boolean expression. Those labels become the column headings, and one gate's output is an input to the next.

6To test a circuit, compare its output column with the intended one. Any row that disagrees exposes the error.

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Worked example

Problem

Build the truth table for (A OR B) AND C.

⚠ Watch out

Reading OR as 'one or the other, but not both' and writing 0 in the row where both inputs are 1. OR gives 1 in that row too. The only 0 is the row where every input is 0.

🧠

Memory hook

OR is generous: one 1 is enough. So don't hunt for the 1s. Hunt for the lone 0, the row where everything is 0.

✓

Check yourself

Without writing out the whole table, say which row of A OR B OR C gives 0, and explain why every other row must give 1.

Flashcards

(13)

What is the OR rule?

The output is 1 if either one or both inputs is 1. It's 0 only when all the inputs are 0.

What is a truth table?

A table showing the outputs for all possible combinations of inputs into a logic gate or logic circuit.

What is a Boolean expression?

An expression that evaluates to either true or false.

In a truth table, what do 1 and 0 stand for?

1 means true; 0 means false.

How many rows does a truth table need for two inputs? For three?

Four rows for two inputs (00, 01, 10, 11). Eight rows for three.

How do you make sure no combination of inputs is missed?

Write the rows as binary numbers in order: 000, 001, 010, 011 and so on.

In a Boolean expression, what is evaluated first?

Brackets. NOT is evaluated before AND and OR.

What does NOT A OR B mean?

(NOT A) OR B. The NOT applies to A only, because NOT is evaluated before OR.

How do you evaluate NOT (A OR B)?

Brackets first: work out the A OR B column, then invert that whole column.

Does the order matter when you evaluate A OR B OR C?

No. With only ORs you can work out B OR C first and then combine it with A. The output column is the same.

What is A OR (A AND B) the same as?

A. It's true when A is true, whether B is true or false.

Building a truth table from a circuit: what are the column headings?

The Boolean expression each gate outputs. The output of one gate becomes an input to the next.

How do you use a truth table to test a circuit?

Compare the circuit's output column with the intended one. Any row that differs exposes an error.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Hunt for the zero

Where does the 11 row go?

Build A OR (B AND C)

Where did this table go wrong?

Does the circuit do what was intended?

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