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Discrete Structures

Chapter 1: Statements, Symbolic, Representation, & Tautologies

QuestionAnswer
Statement/Proposition A sentence that’s T/1 or F/0 - Statement Letters: Represents Statements (Beginning Letters: ABC) - Ex: There are life forms in some planets. - Not 'He is tall' since we don’t know who is ‘he’
Relational Operators - <, >, >=, <= - ==: Equals | Take A’s answer & flip -!=: Not equals
Logical Operator: || Or - Need only 1 right (3 T & 1 F) - ∨: A Disjunction - A & B: Disjuncts of this expression - Ex: A ∨ B
Logical Operator: && And - Need both right (3 F & 1 T) - ∧: A Conjunction - A & B: Conjuncts of this expression - Ex: A ∧ B - Eng Words: And, but, also, more over
Logical Operator: Negation` Not A - Ex: A` - Eng Words: It’s false that A
Logical Operator: Implication → | A (Antecedent) implies B (Consequent) - Ex: A → B - Represents: If statement A then statement B (If study, get good grades) - Only F when A = T & B = F | Same as A` ∨ B - Eng Words: If A, then B | A implies B | A only if B
Logical Operator: Equivalence ←→ - Ex: A ←→ B - Shortcut for (A → B) ∧ (B → A) - Only T when both A & B are T - Eng Words: A if & only if B | A is necessary & sufficient for B
Determining T/F Combos for N Expressions Use 2^N - Ex: 3 Expressions: 2^3 = 8
Order/ Precedence for Logical Connectives 1. Innermost Parentheses 2. ` 3. ∨, ∧ 4. → 5. ←→
Well-Formed Formulas (wff) An expression that’s a legitimate string - Ending Letters to represent wff: PQRS - Main Connective: Last connective to be applied | Solved last in TT - Ex: (A ∨ B) → (B ∧ A) | → MC
Tautology vs Contradiction wff Tautology: A wff that’s always T - Ex: A ∨ A` Contradiction: A wff that’s always F - Ex: A ∧ A`
Equivalent wff Statement P ←→ Statement Q = (P → Q) ∧ (Q → P) - Ex: If P ←→ Q are both wff & a tautology, P can replace Q & vice versa - P & Q have same values = T - Notation: P ⇔ Q - 5 Properties & De Morgan’s Laws
- Difference between P←→Q & P ⇔ Q :
Logical Connective Exercises BLANK
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Commutative Properties ∨ = + & ∧ = * - 1a. A ∨ B ⇔ B ∨ A - 1Ma: A+B = B+ A - 1b. A ∧ B ⇔ B ∧ A - 1Mb: A*B = B*A
Associative Properties - 2a: (A ∨ B) ∨ C ⇔ A ∨ (B ∨ C) - 2Ma: (A+B) + C = A + (B+C) - 2b: (A ∧ B) ∧ C ⇔ A ∧ (B ∧ C) - 2Mb: (A*B) * C = A * (B*C)
Distributive Properties - 3a: A ∨ (B ∧ C) ⇔ (A ∨ B) ∧ (B ∨ C) - 3Ma: A or B & A or C - 3b: A ∧ (B ∨ C) ⇔ (A ∧ B) ∨ (A ∧ C) - 3Mb: A&B or A&C | AB + AC
Identity Properties - 4a: A ∨ 0 ⇔ A - 4Ma: A + 0/F = A - 4b: A ∧ 1 ⇔ A - 4Mb: A * 1/T = A
Complement Properties - 5a: A ∨ A` ⇔ 1 - 5Ma: A or A` = 1 - 5b: A ∧ A` ⇔ 0 - 5Mb: A & A` = 0
De Morgan’s Laws Expresses negation` of a compound statement - (A ∨ B)` ⇔ A` ∧ B` | (A or B)` = A` & B` - (A ∧ B)` ⇔ A` ∨ B` | (A & B)` = A` or B` - Remember: Distribute ` & flip the connective
Created by: Whateverisfree
 

 



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