"Tea or coffee?" Would a computer's OR let you have both?

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.
• Write it as NOT A.

AND

• AND has two inputs. The output is 1 only when both inputs are 1.
• If either input is 0, the output is 0.
• Write it as A AND B.

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".
• Write it as A OR B.

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.
• Cambridge also calls it EOR.
• Write it as A XOR B.

Logic gate symbols

• A logic gate is a circuit that carries out one logical operator.
• AND: flat back, rounded front, two inputs.
• OR: curved back, pointed front, two inputs.
• NOT: a triangle with a small circle at the output. The circle means "reverse".
• XOR: the OR shape with an extra curved line behind the inputs.
• Inputs enter on the left; the output leaves on the right.

Three sensors, one alarm: how many cases must you test?

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.
• 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

Every combination of three inputs

• Each input can be 0 or 1, so each new input doubles the number of rows.
• 1 input: 2 rows. 2 inputs: 4 rows. 3 inputs: 8 rows.
• Fill the input columns by counting up in binary from 000 to 111. Then no row is missed or repeated.
• Exam questions use up to three inputs.

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.

From a problem to a logic statement

• Give each condition a letter (1 = true, 0 = false).
• "and" → AND; "or" / "either" → OR; "not" / "is off" → NOT.
• 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 AND S) OR P

Logic circuits

• A logic circuit joins gates with wires, inputs on the left. Each gate's output feeds the next gate.
• To write the expression, work from the inputs to the output, one gate at a time.
• The alarm X = (D AND S) OR P is an AND gate (inputs D, S) feeding an OR gate with P.
• Draw the statement as given, without simplifying. Every gate except NOT has two inputs.

NAND flash is named after a logic gate. What does that gate do?

NAND

• NAND means NOT AND: an AND gate followed by NOT.
• The output is 0 only when both inputs are 1. Otherwise it is 1.
• Symbol: the AND shape with a small circle (bubble) on the output.
• Write it as A NAND B, which equals NOT (A AND B).

NOR

• NOR means NOT OR: an OR gate followed by NOT.
• The output is 1 only when both inputs are 0. Otherwise it is 0.
• Symbol: the OR shape with a small circle on the output.
• Write it as A NOR B, which equals NOT (A OR B).

All six gates side by side

• A bubble on the output reverses the column: NAND is AND reversed, NOR is OR reversed.
• AND 0001 · NAND 1110 · OR 0111 · NOR 1000 · XOR 0110 (inputs 00, 01, 10, 11).
• NOT is the only gate with one input. All the others have exactly two.
• To spot a gate from its column, count the 1s and see which rows they are on.

From a truth table to an expression

• Find each row where the output is 1.
• For each such row, AND the inputs, with NOT on any input that is 0.
• Join the row terms with OR.
• Example: X = 1 on rows 01 and 10 gives X = (NOT A AND B) OR (A AND NOT B). That is XOR.
• For a circuit, draw one 2-input gate per operator, without simplifying.

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".
• In a circuit, each gate's output feeds the next gate; read it from inputs to output.
• NAND = NOT AND (column 1110). NOR = NOT OR (column 1000). The bubble means NOT.
• To write an expression from a truth table, AND the inputs for each row that outputs 1, then OR the row terms together.