Definitions [5]
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: Logical Equivalence
If two statement patterns have the same truth values in the respective columns of their joint truth table, then these two statement patterns are logically equivalent.
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: Logical Connectives
| Type of compound statement | Connective | Symbol | Example | Set Theory Relation |
|---|---|---|---|---|
| Conjunction | and | ∧ | p and q : p ∧ q | Intersection of sets |
| Disjunction | or | ∨ | p or q : p ∨ q | Union of sets |
| Negation | not | ~ | negation p : ~ p | Complement of a set |
| Conditional or Implication | if...then | → or ⇒ | If p, then q : p → q | Subset relation |
| Biconditional or Double implication | if and only if, i.e., iff | ↔ or ⇔ | p iff q : p ↔ q | Equality of sets |
Key Points: Negation
| Statement | Negation |
|---|---|
| \[\sim(\sim p)\] | ( p ) |
| \[\sim(p\wedge q)\] | \[\sim p\lor q\] |
| \[\sim(p\lor q)\] | \[\sim p\wedge\sim q\] |
| \[\sim(p\to q)\] | \[p\wedge\sim q\] |
| \[\sim(p\leftrightarrow q)\] | \[(p\wedge\sim q)\vee(\sim p\wedge q)\] |
| \[\sim(\forall x)\] | \[\exists x\] |
| \[\sim(\exists x)\] | \[\forall x\] |
| \[\sim(x<y)\] | \[x\geq y\] |
| \[\sim(x>y)\] | \[x\leq y\] |
Key Point: Converse, Inverse, and Contrapositive
| Statement Type | Form | Key Point | Equivalence |
|---|---|---|---|
| Original Statement | \[p\to q\] | If p then q | Equivalent to Contrapositive |
| Converse | \[q\to p\] | Interchange p and q | Equivalent to Inverse |
| Inverse | \[\sim p\to\sim q\] | Negate both p and q | Equivalent to Converse |
| Contrapositive | \[\sim q\to\sim p\] | Interchange + negate | Equivalent to Original Statement |
Key Points: Logical Equivalence
| Concept | Key Point |
|---|---|
| Symbol |
≡ |
| Contrapositive Rule | \[p\to q\equiv\sim q\to\sim p\] |
| Converse & Inverse | \[q\rightarrow p\equiv\sim p\to\sim q\] |
| Important Idea | Equivalent statements give the same result in the truth table |
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 | \[p\leftrightarrow q\equiv(p\to q)\land(q\to p)\]\[\equiv(\sim p\lor q)\land(\sim q\lor p)\] |
Key Points: Application of Logic to Switching Circuits
| Concept | Key Point |
|---|---|
| Series Connection (AND) | Current flows only when both switches are ON |
| Logic Form | \[p\wedge q\] |
| Parallel Connection (OR) | Current flows when any one or both switches are ON |
| Logic Form | \[p\vee q\] |
| Switch ON | Represented by p |
| Switch OFF | Represented by ~p |
| Complementary Switches | If one is ON, the other is OFF → \[S_1\equiv\sim S_2\] |
| Composite Circuits | Combine AND & OR operations (mixed circuits) |
Concepts [9]
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Logical Equivalence
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Algebra of Statements
- Application of Logic to Switching Circuits
