1. The document outlines a lecture on the laws of algebra of propositions, including prerequisites, examples of applications, and the laws themselves. 2. It discusses 13 laws such as the law of double negation, associative laws, distributive laws, and De Morgan's laws. 3. Examples are provided to prove some laws using truth tables and others are proved without tables through the use of other logical equivalences and laws.

1. Program: B.Tech, CSE, 3rd sem/2nd year
CS-302: Discrete Structures
Ms. Neha Sharma
Assistant Professor, Dept. of Mathematics
Unit 3
Logic and Algebra of Propositions
Topic: Laws of Algebra of Propositions
Jul-Dec 2021-22
Lecture 4

2. Outlines
• Prerequisites
• Real Life Examples
• Laws
• Truth Tables of Some Laws
• Examples
• Learning Outcomes
• References
CS-302
2

3. Prerequisites
Basic Mathematical operations
Propositions
Logical Connectives
Well Formed Formulas
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4. Real Life Examples
To design computer circuits.
To construct computer programs.
To verify the correctness of programs.
To build expert systems.
To analyze and solve many familiar puzzles.
Reference No.: R3
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5. Laws on Algebra of Propositions
There are some predefined equivalences and those are as follows:
Let P, Q, and R be statements. Let t be a formula that is a tautology
and let f be a formula that is a contradiction.
Then:
1. The Law of Double Negation: ¬¬P⇔P.
2. The Associative Law for Disjunction: P∨(Q∨R)⇔(P∨Q)∨R.
3. The Associative Law for Conjunction: P∧(Q∧R)⇔(P∧Q)∧R.
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6. Examples
4. The Commutative Law for Disjunction: P∨Q⇔Q∨P.
5. The Commutative Law for Conjunction : P∧Q⇔Q∧P.
6. The First Distributive Law: P∨(Q∧R)⇔(P∨Q)∧(P∨R).
7. The Second Distributive Law: P∧(Q∨R)⇔(P∧Q)∨(P∧R).
8. The Idempotent Law for Disjunction: P∨P⇔P.
9. The Idempotent Law for Conjunction : P∧P⇔P.
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7. Examples Contd.
10. The Identity Law for Tautologies: P∧t⇔P.
11. The Identity Law for Contradictions: P∨f⇔P.
12. The Inverse Law for Tautologies: P∨¬P⇔t.
13. The Inverse Law for Contradictions: P∧¬P⇔f.
14. The Domination Law for Tautologies: P∨t⇔t.
15. The Domination Law for Contradictions: P∧f⇔f.
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8. Laws Contd.
16. De Morgan's Law 1: ¬(P∨Q)⇔¬P∧¬Q.
17. De Morgan's Law 2: ¬(P∧Q)⇔¬P∨¬Q.
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9. Proof of Some Laws with Truth Tables
Commutative Law: 1. P∧Q⇔Q∧P.
2. P∨Q⇔Q∨P.
Similarly the result can be proved for 2 also.
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p q p∧q q∧p p∧q ⇔q∧p
T T T T T
T F F F T
F T F F T
F F F F T

10. Proof of Laws Contd.
Associative Law: 1. P∨(Q∨R)⇔(P∨Q)∨R.
2. P∧(Q∧R)⇔(P∧Q)∧R.
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p q R p∧q q∧r (p∧q)∧r p∧(q∧r) (p∧q)∧r ⇔ p∧(q∧r)
T T T T T T T T
T T F T F F F T
T F T F F F F T
T F F F F F F T
F T T F T F F T
F T F F F F F T
F F T F F F F T
F F F F F F F T

11. Identity Laws
Identity Laws: P∧t⇔P.
P∨f⇔P.
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p t p∧t p∧t ⇔ p
T T T T
F T F T
p f p∨f p∨f ⇔ p
T F T T
F F F T

12. Proving Equivalence without Truth Tables
Question: (∼p ∧(∼q∧r))∨(q∧r)∨(p∧r) ⇔ r, without using
truth tables.
Solution: Consider,
(∼p ∧(∼q∧r))∨(q∧r)∨(p∧r)
⇔ (∼p ∧(∼q∧r))∨((q∨p) ∧r) (distributive law)
⇔ ((∼p ∧∼q)∧r)∨((q∨p) ∧r) (Associative law)
⇔ ((∼p ∧∼q) ∨ (q∨p)) ∧r (Distributive Law)
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13. Proving Equivalence without Truth Tables
⇔ (∼ (p ∨ q) ∨ (q∨p)) ∧ r (Demorgan’s Law)
⇔ T ∧ r (Negation Law)
⇔ r (Identity Law)
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14. Learning Outcomes
Concept of Laws on Algebra of Propositions is learnt.
Questions related to it have also been discussed.
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15. References
1. Dr. D. K. Jain, “Discrete Mathematics”.
2. C.L.Liu, “Elements of Discrete Mathematics”.
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16. CS-302 16