3 missions. Guess, learn, then check. Your place is saved.
Mission 1"Tea or coffee?" Would a computer's OR let you have both?
Mission 2Three sensors, one alarm: how many cases must you test?
Mission 3Is A OR (A AND B) ever different from plain A?
The specification points and sources are listed under the lesson: see Sources.
Mission 1
"Tea or coffee?" Would a computer's OR let you have both?
Make a guess: "Tea or coffee?" Would a computer's OR let you have both?
Boolean logic · AND, OR, NOT
1 or 0, and NOT
A Boolean value has only two states: 1 (TRUE) or 0 (FALSE).
A truth table lists every possible combination of inputs with the output for each.
NOT has one input and reverses it: NOT 0 = 1, NOT 1 = 0.
NOT A is written with an overbar: A̅.
Boolean logic · AND, OR, NOT
AND
AND has two inputs. The output is 1 only when both inputs are 1.
If either input is 0, the output is 0.
A AND B is written A.B
A
B
A AND B
0
0
0
0
1
0
1
0
0
1
1
1
Boolean logic · AND, OR, NOT
OR
OR has two inputs. The output is 1 when at least one input is 1.
The output is 0 only when both inputs are 0.
This is wider than everyday "or", which often means "one or the other, not both".
A OR B is written A + B
A
B
A OR B
0
0
0
0
1
1
1
0
1
1
1
1
Your answer
?A = 1 and B = 0. Work out NOT A, then A AND B, then A OR B. Type the three output bits in that order.
Boolean logic · AND, OR, NOT
XOR
XOR (exclusive OR) has two inputs. The output is 1 when the inputs are different.
If both inputs are the same (both 0 or both 1), the output is 0.
A XOR B is written A ⊕ B
A
B
A XOR B
0
0
0
0
1
1
1
0
1
1
1
0
Quick check
?A hall light has a switch at each end. Flipping either switch changes the light, whatever the other switch is set to. Which operator gives this behaviour?
Quick check
?A cash machine gives out money only if the card is valid (V) and the PIN is correct (P). Which logic statement matches?
Mission check
?"Tea or coffee?" A customer ticks both, so Tea = 1 and Coffee = 1. What does Tea OR Coffee give?
Yes. A computer's OR gives 1 when at least one input is 1, and that includes when both are 1. The output is 0 only when both inputs are 0.
Mission 2
Three sensors, one alarm: how many cases must you test?
Make a guess: Three sensors, one alarm: how many cases must you test?
Boolean logic · combining operators
Combining operators
One logic statement (expression) can use several operators, e.g. (A AND B) OR NOT C.
Work out brackets first.
NOT applies only to the input or bracket just after it.
A.B + C̅ means (A AND B) OR (NOT C): AND is done before OR, like × before +.
Example: (1 AND 0) OR (1 AND 1) = 0 OR 1 = 1
XOR combines in the same way: NOT (1 XOR 0) = NOT 1 = 0
Boolean logic · combining operators
Every combination of three inputs
Each input can be 0 or 1, so each new input doubles the number of rows.
Fill the input columns by counting up in binary from 000 to 111. Then no row is missed or repeated.
Boolean logic · combining operators
Working columns
For a long statement, add a column for each part.
Example: P = (A AND B) OR NOT C.
Column 4: A AND B. Column 5: NOT C. Column 6: P = column 4 OR column 5.
Working columns are optional, but they stop you mixing up steps.
A
B
C
A AND B
NOT C
P
0
0
0
0
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
1
1
0
0
0
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
1
1
1
1
1
1
1
0
1
Your answer
?Q = (A + B).C̅, which means (A OR B) AND NOT C. Complete the output column for the rows 000, 001, 010, 011, 100, 101, 110, 111 (in that order). Type the 8 bits.
Boolean logic · combining operators
From a problem to a logic statement
Give each condition a letter (1 = true, 0 = false).
Use brackets to group conditions that go together.
Example: the alarm X sounds if the door is open (D) and the system is armed (S), or if the panic button (P) is pressed.
X = D.S + P, i.e. X = (D AND S) OR P
Quick check
?A greenhouse sprinkler W switches on when the soil is dry (D) and it is not raining (R), or when the manual override (M) is pressed. Which logic statement matches?
Mission check
?Three sensors, one alarm. How many different input combinations must the truth table cover?
8. Each sensor can be 0 or 1, so each one doubles the number of cases: 2 × 2 × 2 = 8. Counting in binary from 000 to 111 lists all 8 rows, so no case is missed.
Mission 3
Is A OR (A AND B) ever different from plain A?
Make a guess: Is A OR (A AND B) ever different from plain A?
Boolean logic · simplifying expressions
Boolean algebra notation
Eduqas writes Boolean expressions with symbols:
AND: A.B · OR: A + B · NOT: A̅ (a line over the letter) · XOR: A ⊕ B
1 means TRUE and 0 means FALSE, so 1 and 0 can appear in expressions too.
Simplifying means finding a shorter expression with the same truth table.
Boolean logic · simplifying expressions
Equal means the same column
Two expressions are equal if their outputs match on every row of the truth table.
Example: A + A̅. If A = 0, it is 0 + 1 = 1. If A = 1, it is 1 + 0 = 1.
The output is always 1, so A + A̅ = 1.
A truth table is a good way to check a simplification.
Boolean logic · simplifying expressions
Rules with 0, 1 and A itself
Each rule has an OR form and an AND form.
Rule
OR form
AND form
Annulment
A + 1 = 1
A.0 = 0
Identity
A + 0 = A
A.1 = A
Complement
A + A̅ = 1
A.A̅ = 0
Idempotent
A + A = A
A.A = A
Quick check
?Simplify (B + B̅).C
Boolean logic · simplifying expressions
Rules that rearrange
Commutative: order does not matter. A + B = B + A and A.B = B.A
Association: grouping does not matter. A + (B + C) = A + B + C and A.(B.C) = A.B.C
Distribution: multiply out brackets. A.(B + C) = A.B + A.C and A + (B.C) = (A + B).(A + C)
Absorption: A + (A.B) = A and A.(A + B) = A
Boolean logic · simplifying expressions
Simplify step by step
Simplify A.B + A.B̅ and name each rule.
A.B + A.B̅ = A.(B + B̅) — Distribution (taking A out of both terms)
= A.1 — Complement
= A — Identity
Write one step per line and name the rule, as you would in the exam.
Your answer
?Simplify (A + 0).(B + B̅) as far as possible.
Quick check
?Which rule turns A.(B + C) into A.B + A.C?
Mission check
?Is A + A.B ever different from plain A?
Never. If A = 1, A + anything = 1. If A = 0, A.B = 0 as well, so the result is 0. Either way the result equals A: this is the Absorption rule, A + A.B = A.
Recap
Big ideas
Every input and output is 1 (TRUE) or 0 (FALSE). A truth table lists every combination of inputs.
AND: 1 only if both inputs are 1. OR: 1 if at least one input is 1. NOT: reverses its one input.
XOR: 1 only when the two inputs are different.
Work out brackets first; NOT applies only to what comes straight after it.
3 inputs give 8 rows: count in binary from 000 to 111, and add working columns for each part.
Turn a problem into a statement by giving each condition a letter and mapping "and", "or", "not".
Simplifying keeps the truth table the same but uses fewer operations; check with a truth table.
Learn the Appendix B rules: Annulment, Identity, Complement, Idempotent, Commutative, Association, Distribution, Absorption.
Show one step per line and name the rule used.
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Mini-lesson complete!
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You've worked through Logical operations for Eduqas GCSE Computer Science. 🎉
Mission 1"Tea or coffee?" Would a computer's OR let you have both?
Mission 2Three sensors, one alarm: how many cases must you test?
Mission 3Is A OR (A AND B) ever different from plain A?