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Binary Addition, Half Adders & Full Adders

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Denary Addition

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Denary Addition

	4	5	7	1
+		4	5	6

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Denary Addition

	4	5	7	1
+		4	5	6

A

B

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Denary Addition

	4	5	7	1
+		4	5	6
				7

Sum

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Denary Addition

	4	5	7	1
+		4	5	6
				7

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Denary Addition

	4	5	7	1
+		4	5	6
				7

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Denary Addition

	4	5	7	1
+		4	5	6
			2	7

		1

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Denary Addition

	4	5	7	1
+		4	5	6
			2	7

		1

Carry Out

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Denary Addition

	4	5	7	1
+		4	5	6
			2	7

		1

Carry In

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Denary Addition

	4	5	7	1
+		4	5	6
			2	7

		1

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Denary Addition

	4	5	7	1
+		4	5	6
		10	2	7

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Denary Addition

	4	5	7	1
+		4	5	6
		10	2	7

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Denary Addition

	4	5	7	1
+		4	5	6
		0	2	7

	1

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Denary Addition

	4	5	7	1
+		4	5	6
		0	2	7

	1

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Denary Addition

	4	5	7	1
+		4	5	6
	5	0	2	7

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Binary Addition

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Binary Addition

	0	1	1	1
+		1	1	0

A

B

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Binary Addition

	0	1	1	1
+		1	1	0

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Binary Addition

	0	1	1	1
+		1	1	0
				1

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Binary Addition

	0	1	1	1
+		1	1	0
				1

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Binary Addition

	0	1	1	1
+		1	1	0
			2	1

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Binary Addition

	0	1	1	1
+		1	1	0
			10	1

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Binary Addition

	0	1	1	1
+		1	1	0
			10	1

Carry Out

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Binary Addition

	0	1	1	1
+		1	1	0
			0	1

		1

Carry In

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Binary Addition

	0	1	1	1
+		1	1	0
			0	1

		1

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Binary Addition

	0	1	1	1
+		1	1	0
		3	0	1

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Binary Addition

	0	1	1	1
+		1	1	0
		11	0	1

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Binary Addition

	0	1	1	1
+		1	1	0
		11	0	1

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Binary Addition

	0	1	1	1
+		1	1	0
		1	0	1

	1

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Binary Addition

	0	1	1	1
+		1	1	0
		1	0	1

	1

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Binary Addition

	0	1	1	1
+		1	1	0
	1	1	0	1

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Review

0 + 0 = (Sum = 0, Carry = 0)

0 + 1 = 1 (Sum = 1, Carry = 0)

1 + 0 = 1 (Sum = 1, Carry = 0)

1 + 1 = 10 (Sum = 0, Carry = 1)

1 + 1 + 1 = 11 (Sum = 1, Carry = 1)

Right Column 2 inputs (A + B)

Other columns 3 inputs (A + B + Carry In)

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How to implement using logic circuits

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First Column (half adder)

	0	1	1	1
+		1	1	0

A
B
Sum
Carry Out

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First Column (half adder)

	0	1	1	1
+		1	1	0

A
B
Sum
Carry Out

1

0

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First Column (half adder)

A	B	Sum	Carry Out
0	0	0	0
1	0	1	0
0	1	1	0
1	1	0	1

XOR

AND

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Either A or B is 1 (but not both!)

A	B	Sum	Carry Out
0	0	0	0
1	0	1	0
0	1	1	0
1	1	0	1

XOR

AND

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If A and B are 1 then carry one

A	B	Sum	Carry Out
0	0	0	0
1	0	1	0
0	1	1	0
1	1	0	1

XOR

AND

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Second column (Full Adder)

	0	1	1	1
+		1	1	0

A
B
Sum
Carry In Carry Out

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Second column (Full Adder)

	0	1	1	1
+		1	1	0

A
B
Sum
Carry In Carry Out

0

1

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Full Adder

A	B	C in	Sum	C out
0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	0	1
1	0	0	1	0
1	0	1	0	1
1	1	0	0	1
1	1	1	1	1

AND

XOR

OR

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Sum (Output 1 if one / all three inputs are 1s)

A	B	C in	Sum	C out
0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	0	1
1	0	0	1	0
1	0	1	0	1
1	1	0	0	1
1	1	1	1	1

AND

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Carry (Output 1 if at least two inputs are 1s)

A	B	C in	Sum	C out
0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	0	1
1	0	0	1	0
1	0	1	0	1
1	1	0	0	1
1	1	1	1	1

AND