Definitions [4]
Definition: Statements and Truth Values
A statement is a declarative sentence which is either true or false, but not both simultaneously.
- Statements are denoted by lower-case letters p, q, r, etc.
- The truth value of a statement is denoted by ‘1’ or ‘T’ for True and ‘0’ or ‘F’ for False.
- Open sentences, imperative sentences, exclamatory sentences and interrogative sentences are not considered statements in Logic.
Definition: Quantifiers
A quantifier is a symbol used to specify the quantity of elements in a domain for which a statement is true.
- Universal Quantifier (∀): “for all” or “for every”
- Existential Quantifier (∃): “there exists at least one”
Definition: Quantified Statement
A quantified statement is an open statement that becomes a definite statement when a quantifier is applied to it.
Definition: Duality
Two formulae A and B are said to be duals of each other, if either one can be obtained from the other by replacing ∧ by ∨ and ∨ by ∧.
- Replace AND (∧) with OR (∨) and OR (∨) with AND (∧)
Key Points
Key Points: Types of Statement
| Type | Definition |
|---|---|
| Simple Statement | Cannot be broken into smaller statements |
| Compound Statement | Formed by combining two or more simple statements |
| Open Statement | Contains variables; becomes a statement when values are assigned |
Key Points: Types of Statements
| Type | Key Point |
|---|---|
| Tautology | Statement always true |
| Contradiction (Fallacy) | Statement always false |
| Contingency | A statement is sometimes true, sometimes false |
Key Points: Algebra of Statements
| Law | Statement(s) |
|---|---|
| Idempotent Law | \[\begin{array} {l}p\lor p\equiv p \\ p\land p\equiv p \end{array}\] |
| Commutative Law | \[\begin{aligned} & p\lor q\equiv q\lor p \\ & p\land q\equiv q\land p \end{aligned}\] |
| Associative Law | \[(p\lor q)\lor r\equiv p\lor(q\lor r)\equiv p\lor q\lor r\] \[(p\land q)\land r\equiv p\land(q\land r)\equiv p\land q\land r\] |
| Distributive Law | \[p\lor(q\land r)\equiv(p\lor q)\land(p\lor r)\] \[p\land(q\lor r)\equiv(p\land q)\lor(p\land r)\] |
| Identity Law | \[p\lor F\equiv p\] \[p\wedge T\equiv p\] \[p\lor T\equiv T\] \[p\wedge F\equiv F\] |
| Complement Law | \[\begin{array} {l}p\lor\sim p\equiv T \\ p\land\sim p\equiv F \end{array}\] |
| Absorption Law | \[\begin{array} {l}p\lor(p\land q)\equiv p \\ p\land(p\lor q)\equiv p \end{array}\] |
| De Morgan’s Law | \[\sim(p\lor q)\equiv\sim p\land\sim q\] \[\sim(p\wedge q)\equiv\sim p\vee\sim q\] |
| Conditional Law | \[p\to q\equiv\sim p\lor q\] |
| Biconditional Law | \[p\leftrightarrow q\equiv(p\to q)\land(q\to p)\]\[\equiv(\sim p\lor q)\land(\sim q\lor p)\] |
Concepts [12]
- Mathematical Reasoning
- Introduction of Validating Statements
- Mathematically Acceptable Statements
- Statements and Truth Values in Mathematical Logic
- Tautology, Contradiction, and Contingency
- Logical Connective
- Truth Tables
- Logical Equivalance
- Quantifier, Quantified and Duality Statements in Logic
- Converse, Inverse and Contrapositive of the Conditional Staternent
- Negative of a Compound Statement
- Algebra of Statements
