An introductory understanding to the concept of "Propositions" in Discrete Mathematics. It also includes discussion on propositional logic and propositional equivalences

1. Discrete
Mathematics
Lecture 01
Propositions

2. Overview
• Propositions
• Propositional Logic
• Applications of Propositional Logic
• Propositional Equivalences
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3. Propositions
Understanding the Concept of
“Propositions”
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4. Propositions
• Rules of logic specify mathematical statements
• There exists and integer not the sum of two squares
• For every positive integer 𝑛𝑛, the sum of positive
integers not exceeding 𝑛𝑛 is
𝑛𝑛 𝑛𝑛+1
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• A proposition is a declarative sentence
• Either True or False
• Not both
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5. Propositions
• Washington, D.C., is the capital of U.S.A.
• Toronto is the capital of Canada
• 1 + 1 = 2
• 2 + 2 = 3
• What time is it?
• Read this carefully
• 𝑥𝑥 + 1 = 2
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6. Propositions
• Propositions are expressed through variables
• Makes operations easier
• Letters denote propositional variables
• Convention: 𝑝𝑝, 𝑞𝑞, 𝑟𝑟, 𝑠𝑠, …
• They are either true or false
• Truth values denoted by T; when proposition is true
• Truth values denoted by F; when proposition is false
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7. Propositional
Logic
Discussing different types of
propositional logic
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8. Propositional Logic
• Compound Propositions
• New propositions
• Formed from existing propositions
• Through logical operators
• Connectives
• Some of the logical operators
• Form propositions using multiple propositions
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9. Propositional Logic
• Negation
• The negation of 𝑝𝑝 is denoted by ¬𝑝𝑝 (or ̅
𝑝𝑝)
• Makes: “It is not the case that 𝑝𝑝”
• Read as “not 𝑝𝑝”
• Example
• 𝑝𝑝: Michael’s PC runs Linux
• ¬𝑝𝑝: It is not the case that Michael’s PC runs Linux
• Simply: Michael’s PC does not run Linux
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10. Propositional Logic
• Conjunction
• The conjunction of 𝑝𝑝 and 𝑞𝑞 is denoted by 𝑝𝑝 ∧ 𝑞𝑞
• Read as “𝑝𝑝 and 𝑞𝑞”
• Sometimes “but” instead of “and” → The sun is shining, but it is
raining
• Example
• 𝑝𝑝: Monir’s PC has 512 GB SSD
• 𝑞𝑞: Monir’s PC runs faster than 2 GHz
• 𝑝𝑝 ∧ 𝑞𝑞: Monir’s PC has 512 GB SSD and Monir’s PC runs faster
than 2 GHz
• Simply: Monir’s PC has 512 GB SSD and runs faster than 2 GHz
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11. Propositional Logic
• Disjunction
• The disjunction of 𝑝𝑝 and 𝑞𝑞 is denoted by 𝑝𝑝 ∨ 𝑞𝑞
• Read as “𝑝𝑝 or 𝑞𝑞”
• Two variations: “exclusive or”, “inclusive or”
• Example
• 𝑝𝑝: Monir’s PC has 512 GB SSD
• 𝑞𝑞: Monir’s PC runs faster than 2 GHz
• 𝑝𝑝 ∨ 𝑞𝑞: Monir’s PC has 512 GB SSD or Monir’s PC runs faster than 2 GHz
• Simply: Monir’s PC has 512 GB SSD or runs faster than 2 GHz
• Must contain at least one
• Can contain both (inclusive or) or only one (exclusive or)
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12. Propositional Logic
• Condition Statements
• The conditional statements of 𝑝𝑝 and 𝑞𝑞 is denoted by
𝑝𝑝 → 𝑞𝑞
• Read as “If 𝑝𝑝 then 𝑞𝑞” and many other ways
• Example
• 𝑝𝑝: You get 100% on the final
• 𝑞𝑞: You get an A
• 𝑝𝑝 → 𝑞𝑞: If you get 100% on the final, then you get an A
• Another: If you have a smartphone, then 2 + 3 = 5
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13. Propositional Logic
• Converse
• 𝑞𝑞 → 𝑝𝑝 is called converse of 𝑝𝑝 → 𝑞𝑞
• Contrapositive
• ¬𝑞𝑞 → ¬𝑝𝑝 is called contrapositive of 𝑝𝑝 → 𝑞𝑞
• Always has the same truth value as condition
• Inverse
• ¬𝑝𝑝 → ¬𝑞𝑞 is called inverse of 𝑝𝑝 → 𝑞𝑞
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14. Propositional Logic
• Bidirectional
• The bidirectional of 𝑝𝑝 and 𝑞𝑞 is denoted by 𝑝𝑝 ↔ 𝑞𝑞
• Read as “𝑝𝑝 if and only if 𝑞𝑞”
• Bi-implications
• Similar to 𝑝𝑝 → 𝑞𝑞 ∧ (𝑞𝑞 → 𝑝𝑝)
• Example
• 𝑝𝑝: You can take the flight
• 𝑞𝑞: You buy a ticket
• 𝑝𝑝 ↔ 𝑞𝑞: You can take the if and only if (iff) you buy a ticket
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15. Propositional Logic
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𝒑𝒑 𝒒𝒒 ¬𝒑𝒑 ¬𝒒𝒒 𝒑𝒑 ∧ 𝒒𝒒 𝒑𝒑 ∨ 𝒒𝒒 𝒑𝒑 ⊗ 𝒒𝒒 𝒑𝒑 → 𝒒𝒒 𝒑𝒑 ↔ 𝒒𝒒
T T F F T T F T T
T F F T F T T F F
F T T F F T T T F
F F T T F F F T T

16. Propositional Logic
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𝒑𝒑 𝒒𝒒 ¬𝒒𝒒 𝒑𝒑 ∨ ¬𝒒𝒒 𝒑𝒑 ∧ 𝒒𝒒 (𝒑𝒑 ∨ ¬𝒒𝒒) → (𝒑𝒑 ∧ 𝒒𝒒)
T T F T T T
T F T T F F
F T F F F T
F F T T F F
Truth Tables of Compound Propositions: (𝒑𝒑 ∨ ¬𝒒𝒒) → (𝒑𝒑 ∧ 𝒒𝒒)

17. Propositional Logic
• Precedence of Logical Operators
• ¬ > ∧ > ∨ > → > ↔
• Use brackets for clarification
• Logic and Bit Operations
• Bit: Binary Digit
• 1 can represent True/T
• 0 can represent False/F
• Bit strings, length, Bitwise operations
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18. Applications of
Propositional
Logic
Where propositional logics are
used
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19. Applications of Propositional
Logic
• Translating English sentences
• System specifications
• Boolean searches
• Logic puzzles
• Logic circuits
• Read examples from books to understand
scenario
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20. Propositional
Equivalences
Understanding and using
propositional equivalences
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21. Propositional Equivalences
• Tautology
• A compound proposition that is always true
• Contradiction
• A compound proposition that is always false
• Contingency
• A compound proposition neither true, nor false
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22. Propositional Equivalences
• Logical Equivalences
• Separate compound propositions
• Same truth values in all cases
• Let 𝑝𝑝 and 𝑞𝑞 be two separate compound propositions
• 𝑝𝑝 and 𝑞𝑞 logically equivalent if 𝑝𝑝 ↔ 𝑞𝑞 is a tautology.
• 𝑝𝑝 ≡ 𝑞𝑞 (or 𝑝𝑝 ⇔ 𝑞𝑞) denotes that 𝑝𝑝 and 𝑞𝑞 are logically
equivalent
• 𝑝𝑝 ≡ 𝑞𝑞 is not a compound proposition, but a statement
that 𝑝𝑝 ↔ 𝑞𝑞 is a tautology
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23. Propositional Equivalences
• Logical equivalences proved by truth tables
• Find the true value for each proposition in all
cases
• If the truth values matches for all, they are
equivalent
• Example: ¬(𝑝𝑝 ∨ 𝑞𝑞)and ¬𝑝𝑝 ∧ ¬𝑞𝑞 are logically
equivalent
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24. Propositional Equivalences
• Identity laws
• 𝑝𝑝 ∧ 𝑻𝑻 ≡ 𝑝𝑝 and 𝑝𝑝 ∨ 𝐹𝐹 ≡ 𝑝𝑝
• Domination laws
• 𝑝𝑝 ∨ 𝑻𝑻 ≡ 𝑻𝑻 and 𝑝𝑝 ∧ 𝐹𝐹 ≡ 𝑭𝑭
• Idempotent laws
• 𝑝𝑝 ∨ 𝑝𝑝 ≡ 𝑝𝑝 and 𝑝𝑝 ∧ 𝑝𝑝 ≡ 𝑝𝑝
• Double negation law
• ¬(¬𝑝𝑝) ≡ 𝑝𝑝
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25. Propositional Equivalences
• Commutative laws
• 𝑝𝑝 ∨ 𝑞𝑞 ≡ 𝑞𝑞 ∨ 𝑝𝑝 and 𝑝𝑝 ∧ 𝑞𝑞 ≡ 𝑞𝑞 ∧ 𝑝𝑝
• Associative laws
• (𝑝𝑝 ∨ 𝑞𝑞) ∨ 𝑟𝑟 ≡ 𝑝𝑝 ∨ (𝑞𝑞 ∨ 𝑟𝑟) and (𝑝𝑝 ∧ 𝑞𝑞) ∧ 𝑟𝑟 ≡ 𝑝𝑝 ∧ (𝑞𝑞 ∧ 𝑟𝑟)
• Distributive laws
• 𝑝𝑝 ∨ 𝑞𝑞 ∧ 𝑟𝑟 ≡ 𝑝𝑝 ∨ 𝑞𝑞 ∧ 𝑝𝑝 ∨ 𝑟𝑟
• 𝑝𝑝 ∧ (𝑞𝑞 ∨ 𝑟𝑟) ≡ (𝑝𝑝 ∧ 𝑞𝑞) ∨ (𝑝𝑝 ∧ 𝑟𝑟)
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26. Propositional Equivalences
• De Morgan’s laws
• ¬(𝑝𝑝 ∧ 𝑞𝑞) ≡ ¬𝑝𝑝 ∨ ¬𝑞𝑞 and ¬(𝑝𝑝 ∨ 𝑞𝑞) ≡ ¬𝑝𝑝 ∧ ¬𝑞𝑞
• Absorption laws
• 𝑝𝑝 ∨ (𝑝𝑝 ∧ 𝑞𝑞) ≡ 𝑝𝑝 and 𝑝𝑝 ∧ (𝑝𝑝 ∨ 𝑞𝑞) ≡ 𝑝𝑝
• Negation laws
• 𝑝𝑝 ∨ ¬𝑝𝑝 ≡ 𝑻𝑻 and 𝑝𝑝 ∧ ¬𝑝𝑝 ≡ 𝑭𝑭
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27. Propositional Equivalences
• There exists many such equivalences
• The more you know, the better
• Helps evaluate larger propositions faster
• Faster than truth table
• A notation for long compound proposition
• 𝑝𝑝1 ∧ 𝑝𝑝2 ∧ 𝑝𝑝3 ∧ ⋯ ∧ 𝑝𝑝𝑛𝑛 can be written as ∧𝑗𝑗=1
𝑛𝑛
𝑝𝑝𝑗𝑗
• 𝑝𝑝1 ∨ 𝑝𝑝2 ∨ 𝑝𝑝3 ∨ ⋯ ∨ 𝑝𝑝𝑛𝑛 can be written as ∨𝑗𝑗=1
𝑛𝑛
𝑝𝑝𝑗𝑗
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28. Propositional Equivalences
• De Morgan’s law is particularly important
• Helps a lot in logical equivalences
• Read book example for better understanding
• Constructing new logical equivalences
• Using existing logical equivalences
• Kind of like simplification
• Make new logical equivalence
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29. Propositional Equivalences
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Show that ¬ 𝑝𝑝 ∨ ¬𝑝𝑝 ∧ 𝑞𝑞 ≡ ¬𝑝𝑝 ∧ ¬𝑞𝑞
Solution:
¬ 𝑝𝑝 ∨ ¬𝑝𝑝 ∧ 𝑞𝑞 ≡ ¬𝑝𝑝 ∧ ¬ ¬𝑝𝑝 ∧ 𝑞𝑞 [Second De Morgan’s Law]
≡ ¬𝑝𝑝 ∧ ¬ ¬𝑝𝑝 ∨ ¬𝑞𝑞 [First De Morgan’s Law]
≡ ¬𝑝𝑝 ∧ 𝑝𝑝 ∨ ¬𝑞𝑞 [Double Negation Law]
≡ ¬𝑝𝑝 ∧ 𝑝𝑝 ∨ ¬𝑝𝑝 ∧ ¬𝑞𝑞 [Second Distributive Law]
≡ 𝑭𝑭 ∨ ¬𝑝𝑝 ∧ ¬𝑞𝑞 [Because, 𝑝𝑝 ∧ ¬𝑝𝑝 ≡ 𝑭𝑭]
≡ ¬𝑝𝑝 ∧ 𝑞𝑞 [Identity law for 𝑭𝑭]

30. Propositional Equivalences
• Propositional Satisfiability
• When no combinations exist to make a compound
proposition true, it is unsatisfiable
• In other words, the negation of the compound
proposition is a tautology
• The case where true value of a compound proposition
is true is a solution
• Build a truth table for a compound proposition to prove
whether it is satisfiable or unsatisfiable
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31. Propositional Equivalences
• Applications of Satisfiability
• Sudoku
• Read book for scenario
• Solving satisfiable problems
• For every 𝑥𝑥 variable compound proposition there is 2𝑥𝑥
possible combinations
• Brute force approaches are time consuming
• We need algorithms
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32. Thank You
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