This document discusses Boolean expressions, truth tables, and how to represent logic functions in Sum of Products (SOP) and Product of Sums (POS) form. It begins by defining Boolean expressions and truth tables. It then shows examples of constructing truth tables from logic diagrams and Boolean expressions. Next, it covers maxterms and minterms and represents a logic function as a SOP using these concepts. Finally, it demonstrates how to reduce a SOP to minimal SOP form using Boolean algebra properties and also defines the POS form.

1. Digital Logic Design
By
Jalpa Maheshwari
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2. What is Boolean Expression?
A Boolean expression is a logical statement that is either TRUE or FALSE.
What is Truth Table?
 Logic
A diagram in rows and columns showing how the truth or falsity of a proposition varies with
that of its components.
 Electronics
A diagram of the outputs from all possible combinations of input.
 Truth table can be constructed using Boolean expressions and vice versa.
Truth Tables and Boolean Expression
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3. Truth table and Boolean Expression using Logic Diagram:
 Logic Diagram:
 Truth Table:
Truth Tables and Boolean Expression
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Inputs Outputs
A B C = A. B D = A ⊕ B Q=C ⊕ D
0 0 1 0 0
0 1 1 1 1
1 0 1 1 1
1 1 0 0 1

4. Truth Tables and Boolean Expression
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 Boolean Expression
𝐶 = 𝐴. 𝐵
𝐷 = 𝐴 ⊕ 𝐵
𝑄 = 𝐶 ⊕ 𝐷

5. Logic diagram and Truth table using Boolean Expression:
 Boolean Expression:
 Logic Diagram:
Truth Tables and Boolean Expression
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A. B. C + A. (B + C

6. Truth Table:
Truth Tables and Boolean Expression
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Inputs Outputs
A B C A.B.C 𝐁 𝐂 𝐁 + 𝐂 𝐀. ( 𝐁 + 𝐂 𝐀. 𝐁. 𝐂 + 𝐀. ( 𝐁 + 𝐂
0 0 0 0 1 1 1 0 0
0 0 1 0 1 0 1 0 0
0 1 0 0 0 1 1 0 0
0 1 1 0 0 0 0 0 0
1 0 0 0 1 1 1 1 1
1 0 1 0 1 0 1 1 1
1 1 0 0 0 1 1 1 1
1 1 1 1 0 0 0 0 1

7. Maxterm and Minterm:
 Minterms are those terms, where output Y is high. It is represented by ‘m’. The order of
inputs A, B and C does matter.
 Maxterms are those terms, where output Y is low. It is represented by ‘M’.
Minterm and Maxterm
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Inputs Outputs
A B C Y
𝑚0 0 0 0 1
𝑚1 0 0 1 0
𝑚2 0 1 0 1
𝑚3 0 1 1 1
𝑚4 1 0 0 0
𝑚5 1 0 1 0
𝑚6 1 1 0 1
𝑚7 1 1 1 1
𝑦 𝐴, 𝐵, 𝐶 = 𝜋𝑀(1,4,5
 Minterm:
𝑦 𝐴, 𝐵, 𝐶 = Ʃ𝑚(0,2,3,6,7
 Maxterm:

8. Sum of Products and Product of Sums:
Sum of Products:
Convention
A = 0
A = 1
Lets take the previous problem, we want to represent the output ‘Y’ into SOP form.
Take those inputs in considerations, where output ‘Y’ is high.
SOP and POS
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𝑦 = (𝐴. 𝐵. 𝐶 + 𝐴. 𝐵. 𝐶 + 𝐴. 𝐵. 𝐶 + 𝐴. 𝐵. 𝐶 + (𝐴. 𝐵. 𝐶
This form is known as Canonical SOP (CSOP) Form, each
minterm contains actual variable and its complement form.
So, this equation can be reduced to Minimal SOP (MSOP)
using Boolean Algebra, K-map or any reduction method.

9. Properties
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Boolean Algebra Properties
Commutative a + b = b + a a. b = b. a
Associative 𝑎 + 𝑏 + 𝑐 = 𝑎 + (𝑏 + 𝑐 𝑎. 𝑏 . 𝑐 = 𝑎. (𝑏. 𝑐
Distributive 𝑎. 𝑏 + 𝑐
= 𝑎. 𝑏 + (𝑎. 𝑐
𝑎 + 𝑏. 𝑐
= 𝑎 + 𝑏 . (𝑎 + 𝑐
Idempotent 𝑎 + 𝑎 = 𝑎 𝑎. 𝑎 = 𝑎
Identity 𝑎 + 0 = 𝑎 𝑎. 1 = 𝑎
Annihilate 𝑎 + 1 = 1
𝑎 + 𝑎 = 1
𝑎. 0 = 0
𝑎. 𝑎 = 0
De-Morgan Theorem 𝑎 + 𝑏 = 𝑎. 𝑏 𝑎. 𝑏 = 𝑎 + 𝑏
Inverse 𝑎 = 𝑎

10. CSOP reduction to MSOP:
SOP and POS
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𝑦 = (𝐴. 𝐵. 𝐶 + 𝐴. 𝐵. 𝐶 + 𝐴. 𝐵. 𝐶 + 𝐴. 𝐵. 𝐶 + 𝐴. 𝐵. 𝐶
𝑦 = (𝐴. 𝐵. 𝐶 + 𝐴. 𝐵 𝐶 + 𝐶 + 𝐴. 𝐵( 𝐶 + 𝐶
𝑈𝑠𝑖𝑛𝑔 𝐴𝑛𝑛𝑖ℎ𝑖𝑙𝑎𝑡𝑒 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑎 + 𝑎 = 1
𝑦 = (𝐴. 𝐵. 𝐶 + 𝐴. 𝐵 + 𝐴. 𝐵
𝑦 = (𝐴. 𝐵. 𝐶 + 𝐵( 𝐴 + 𝐴
𝑦 = (𝐴. 𝐵. 𝐶 + 𝐵
𝑙𝑒𝑡𝑠 𝑠𝑢𝑝𝑝𝑜𝑠𝑒 𝑥 = 𝐴 𝐶
𝑦 = 𝑥. 𝐵 + 𝐵
𝑈𝑠𝑖𝑛𝑔 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑐𝑒 𝑙𝑎𝑤:
𝑦 = 𝑥 + 𝐵
𝑦 = 𝐴 𝐶 + B Minimal SOP

11. Product of Sum:
From previous example
Convention:
A = 1
A = 0
Take inputs terms, where output ‘y’ is low.
SOP and POS
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𝑦 = 𝐴 + 𝐵 + 𝐶 . 𝐴 + 𝐵 + 𝐶 . ( 𝐴 + 𝐵 + 𝐶
Canonical Product of sum

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