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Logic Gates, Half Adder and Full Adder, ISC 11

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Computer ScienceISC Class 11 · Propositional Logic and Logic Gates

Logic Gates, Half Adder and Full Adder (ISC Class 11 Computer Science)

Updated 17 October 2026 · Fact-checked

Logic gates are circuits that apply a logical operation to binary inputs. AND gives 1 only if all inputs are 1; OR gives 1 if any input is 1; NOT inverts; NAND and NOR are the inverted AND and OR; XOR gives 1 when the inputs differ; XNOR gives 1 when they are equal. A half adder adds two bits (Sum = A XOR B, Carry = A AND B); a full adder adds two bits and a carry-in (Sum = A XOR B XOR Cin, Carry-out = AB + Cin(A XOR B)).

Notice the first half adder's sum feeds the second one with Cin, and the two carries meet at the OR gate to make Cout.

Illustration drawn for QuestionBuddy.

Understand Logic Gates, the Half Adder and the Full Adder

Gates and truth tables (inputs A B = 00, 01, 10, 11).

GateExpressionOutput
ANDA . B0 0 0 1
ORA + B0 1 1 1
NOTA'inverts A
NAND(A . B)'1 1 1 0
NOR(A + B)'1 0 0 0
XORA xor B0 1 1 0
XNOR(A xor B)'1 0 0 1

NAND and NOR are universal gates: any circuit can be built from NAND gates alone (NOT: NAND with the inputs joined; AND: NAND followed by NOT; OR: NOT of each input then NAND).

Half adder. Adds two bits A and B. Sum S = A xor B; Carry C = A . B. Truth table (A B: S C): 00: 0 0; 01: 1 0; 10: 1 0; 11: 0 1. It needs one XOR gate and one AND gate. It has no input for a carry from a lower position.

Full adder. Adds A, B and a carry-in Cin. Sum S = A xor B xor Cin; Carry-out Cout = A.B + Cin.(A xor B).

ABCinSumCout
00000
00110
01010
01101
10010
10101
11001
11111

A full adder can be made from two half adders and an OR gate.

Adding binary numbers. Use a half adder for the least significant bit and full adders for the higher bits, passing each carry-out to the next stage.

Logic Gates, the Half Adder and the Full Adder: key facts

Half adder
Sum = A xor B | Carry = A AND B
Two inputs.
Full adder
Sum = A xor B xor Cin | Cout = A.B + Cin.(A xor B)
Three inputs.
Universal gates
NAND and NOR | Any gate can be built from either
Universal.

How to answer Logic Gates, the Half Adder and the Full Adder questions

Evaluate the circuit gate by gate from the inputs.

  1. 1Write the output of each gate in order from the inputs.
  2. 2For an adder, find Sum and Carry for the bit position.
  3. 3Pass the carry to the next position.
  4. 4Check the result by converting to decimal.

Logic Gates, the Half Adder and the Full Adder in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Half adder

Suppose two switches A and B each give 1 when pressed and a display must show the count of pressed switches in two bits.

What are Sum and Carry for both switches pressed?

Try it first, then show the solutionSolution
  1. A = 1, B = 1: Sum = 1 xor 1 = 0; Carry = 1 AND 1 = 1.
  2. The count shown is 10 in binary, which is 2.

Answer: Sum 0, Carry 1 (binary 10 = 2).

Memory hook: 1 + 1 = 10 in binary.

Tricks to remember Logic Gates, the Half Adder and the Full Adder

Short memory aids. Say each one aloud once, then use it on a question the same day.

  • XOR = different

    XOR gives 1 when the inputs differ.

    Use it when: Sum.

  • AND for carry

    Carry is 1 only when both bits are 1.

    Use it when: Half adder.

  • Full = two half plus OR

    Two half adders and an OR gate.

    Use it when: Full adder.

Logic Gates, the Half Adder and the Full Adder: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Gate · 1 mark · MCQBoard level

The output of a NAND gate for inputs 1 and 1 is:
(a) 1 (b) 0 (c) undefined (d) same as OR

Try it first, then show the solutionSolution
  1. AND of 1 and 1 is 1; NAND inverts it: 0.

Answer: (b) 0

Example 2: Half adder table · 3 marks · Short answerBoard level

Draw the truth table of a half adder and write the Boolean expressions for Sum and Carry. State the gates needed. (3 marks)

Try it first, then show the solutionSolution
  1. A B: Sum, Carry = 00: 0, 0; 01: 1, 0; 10: 1, 0; 11: 0, 1.
  2. Sum = A xor B; Carry = A AND B.
  3. One XOR gate and one AND gate.

Answer:

ABSumCarry
0000
0110
1010
1101

Sum = A xor B; Carry = A.B; gates: XOR and AND.

Example 3: Add two 2-bit numbers · 6 marks · Long answerHarder board level

Add the binary numbers 11 and 01 using a half adder for the least significant bit and a full adder for the next bit. Show the Sum and Carry at each stage and the final result in binary and decimal. (6 marks)

Try it first, then show the solutionSolution
  1. Bit 0: A = 1, B = 1 (half adder): Sum = 0, Carry = 1.
  2. Bit 1: A = 1, B = 0, Cin = 1 (full adder): Sum = 1 xor 0 xor 1 = 0; Cout = 1.0 + 1.(1 xor 0) = 0 + 1 = 1.
  3. The final carry-out is the third bit: result = 1 0 0.
  4. Check: 3 + 1 = 4 = 100 in binary.

Answer: 11 + 01 = 100 (binary) = 4 (decimal); stage 0: Sum 0, Carry 1; stage 1: Sum 0, Carry 1.

Example 4: Trace a circuit · 5 marks · Long answerHarder board level

A circuit gives Y = (A AND B) OR (NOT C). Find Y for (A, B, C) = (1, 1, 0), (1, 0, 1), (0, 0, 0). Then build Y from NAND gates only for two inputs: write NOT A and A AND B using NAND. (5 marks)

Try it first, then show the solutionSolution
  1. (1,1,0): (1 AND 1) = 1; NOT 0 = 1; 1 OR 1 = 1.
  2. (1,0,1): 0 OR 0 = 0.
  3. (0,0,0): 0 OR 1 = 1.
  4. NOT A = A NAND A. A AND B = (A NAND B) NAND (A NAND B).

Answer: Y = 1, 0, 1. NOT A = A NAND A; A AND B = (A NAND B) NAND (A NAND B).

Example 5: Full adder reasoning · 5 marks · Long answerHOTS / competency

For a full adder find Sum and Cout when A = 1, B = 1, Cin = 1, and when A = 0, B = 1, Cin = 1. Explain why a half adder alone cannot add multi-bit numbers, and how many half adders and other gates make one full adder. (5 marks)

Try it first, then show the solutionSolution
  1. 1,1,1: Sum = 1 xor 1 xor 1 = 1; Cout = 1.1 + 1.(0) = 1.
  2. 0,1,1: Sum = 0 xor 1 xor 1 = 0; Cout = 0 + 1.(1) = 1.
  3. A half adder has no carry-in input, so it cannot take a carry from the previous position.
  4. A full adder = 2 half adders + 1 OR gate (the OR combines the two carries).

Answer: 1,1,1: Sum 1, Cout 1; 0,1,1: Sum 0, Cout 1. A half adder has no carry-in; a full adder is two half adders plus an OR gate.

Where students lose marks in Logic Gates, the Half Adder and the Full Adder

The slips examiners see most often, and what to write instead.

  • Mistake: Writing Sum = A AND B

    Mixing sum and carry.

    Wrong working

    Sum = A.B

    Right working

    Sum = A xor B

    Do this: AND is for carry.

    Usually costs: 2 marks.

  • Mistake: Dropping the final carry

    Short result.

    Wrong working

    11 + 01 = 00

    Right working

    100

    Do this: Include the last carry-out.

    Usually costs: 2 marks.

  • Mistake: XNOR for XOR

    Inversion mix-up.

    Wrong working

    XOR gives 1 when equal

    Right working

    XOR gives 1 when different

    Do this: XNOR is the equal gate.

    Usually costs: 1 mark.

Exam tips

  • Draw the truth table before writing expressions.
  • Carry the last carry-out into the result.
  • Check binary addition in decimal.

Practice questions on Logic Gates, the Half Adder and the Full Adder

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

  1. Q1 · 1 mark · MCQBoard level

    The output of an XOR gate is 1 when:
    (a) both inputs are 1 (b) the inputs are different (c) both inputs are 0 (d) always

    Hint

    Exclusive.

    Answer

    (b) the inputs are different

  2. Q2 · 2 marks · Short answerBoard level

    Write the truth table of NOR and XNOR for inputs A and B. (2 marks)

    Hint

    Invert OR and XOR.

    Answer

    Inputs 00, 01, 10, 11: NOR = 1, 0, 0, 0; XNOR = 1, 0, 0, 1.

  3. Q3 · 5 marks · Long answerHarder board level

    Add 101 and 011 using one half adder and two full adders. Show Sum and Carry at each stage and the final result. (5 marks)

    Hint

    5 + 3 = 8.

    Answer

    Bit 0: 1 + 1: Sum 0, Carry 1 (half adder). Bit 1: 0 + 1 + 1: Sum 0, Carry 1. Bit 2: 1 + 0 + 1: Sum 0, Carry 1. Final carry 1: result 1000 = 8 (5 + 3).

  4. Q4 · 4 marks · Short answerHarder board level

    Build a half adder from NAND gates: write Sum and Carry in terms of NAND (hint: A xor B = (A NAND (A NAND B)) NAND (B NAND (A NAND B))). Verify for A = 1, B = 0. (4 marks)

    Hint

    Evaluate step by step.

    Answer

    Let X = A NAND B. Sum = (A NAND X) NAND (B NAND X); Carry = X NAND X. For A = 1, B = 0: X = 1; A NAND X = 0; B NAND X = 1; Sum = 0 NAND 1 = 1. Carry = 1 NAND 1 = 0. This agrees with 1 + 0 = 01.

  5. Q5 · 4 marks · Short answerHOTS / competency

    A security light turns ON (Y = 1) when it is dark (D = 1) and either motion is detected (M = 1) or the manual switch (S = 1) is on. Write Y, draw the gate list, and give Y for (D, M, S) = (1, 0, 1) and (0, 1, 1). (4 marks)

    Hint

    D AND (M OR S).

    Answer

    Y = D . (M + S): one OR gate and one AND gate. For (1, 0, 1): 1 . 1 = 1 (light on). For (0, 1, 1): 0 . 1 = 0 (light off).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

  1. 1.Half adder Sum?Show answer

    A xor B.

  2. 2.Half adder Carry?Show answer

    A AND B.

  3. 3.Universal gates?Show answer

    NAND and NOR.

  4. 4.Full adder inputs?Show answer

    A, B and carry-in.

  5. 5.1 + 1 in binary?Show answer

    10.

Logic Gates, the Half Adder and the Full Adder: frequently asked questions

What is a carry?

The bit passed to the next position.

How many rows in a full adder table?

8.

What is NOT called?

Inverter.

Which gate is XNOR?

Equality gate.