Algebraic Properties of Statements | Advanced Mathematics | Class 9
Algebraic Properties of Statements | Advanced Mathematics | Class 9
Algebraic operations are performed among the statements also, but here the operations are the connectives. Let \(p\), \(q\) and \(r\) be three statements. Let \(T\) denotes some true and \(F\) denotes some false statements then the following equivalent statements are valid. Also, with the help of truth tables all of them are proved as follow-
Idempotent Law
The Idempotent Law in propositional logic states that applying a logical operation (AND or OR) to a statement with itself produces a statement that is equivalent to the original. In other words, if you perform an operation on a statement and then perform the same operation again, the result remains unchanged. This law underscores the idea that repetition of a logical operation does not alter the truth value of a statement.
- \( p\vee p \equiv p\)
| \(\ p \) |
\(\ p\vee p\) |
| \(\ T \) |
\(\ T\) |
| \(\ F \) |
\(\ F\) |
- \( p\wedge p \equiv p\)
| \(\ p \) |
\(\ p\wedge p\) |
| \(\ T \) |
\(\ T\) |
| \(\ F \) |
\(\ F\) |
Associative Law
The Associative Law asserts that the grouping of logical operations does not affect the overall truth value of a compound statement. This means that when you have multiple operations of the same type (AND or OR), you can regroup them without altering the truth value. Essentially, it emphasizes the independence of grouping when it comes to logical operations.
- \((p\vee q)\vee r\equiv p\vee (q\vee r) \)
| \(\ p \) |
\(\ q \) |
\(\ r \) |
\(\ p\vee{q} \) |
\(\ q\vee{r} \) |
\(\ (p\vee{q})\vee{r} \) |
\(\ p\vee{({q}\vee{r})}\) |
| \(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
| \(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
- \((p\wedge q)\wedge r\equiv p\wedge (q\wedge r)\)
| \(\ p \) |
\(\ q \) |
\(\ r \) |
\(\ p\wedge{q} \) |
\(\ q\wedge{r} \) |
\(\ (p\wedge{q})\wedge{r} \) |
\(\ p\wedge{({q}\wedge{r})}\) |
| \(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
Commutative Law
The Commutative Law states that the order of operands does not affect the outcome of certain logical operations (specifically AND and OR). For instance, in an AND operation, the truth value remains the same whether the operands are switched or not. Similarly, in an OR operation, the truth value remains unaffected by changing the order of the operands. This property highlights the symmetry in certain logical operations.
- \(p\vee q \equiv q\vee p\)
| \(\ p \) |
\(\ q \) |
\(\ p \vee{q} \) |
\(\ q \vee{p} \) |
| \(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
| \(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T\) |
| \(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F\) |
- \(p\wedge q\equiv q\wedge p\)
| \(\ p \) |
\(\ q \) |
\(\ p \wedge{q} \) |
\(\ q \wedge{p} \) |
| \(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F\) |
| \(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F\) |
Distributive Law
The Distributive Law is a fundamental property in logic that governs the interaction between logical operations, usually AND and OR. It states that you can distribute an operation over another operation without changing the truth value of the overall expression. In essence, it allows you to break down complex logical expressions into simpler ones, facilitating analysis and manipulation.
- \(p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r) \)
| \(\ p \) |
\(\ q \) |
\(\ r \) |
\(\ q\wedge{r} \) |
\(\ p\vee{q} \) |
\(\ p\vee{r} \) |
\(\ p\vee{(q\wedge{r})} \) |
\(\ (p\vee{q})\wedge{(p\vee{r})} \) |
| \(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ F \) |
\(\ T \) |
\(\ F\) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
- \( p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)\)
| \(\ p \) |
\(\ q \) |
\(\ r \) |
\(\ q\vee{r} \) |
\(\ p\wedge{q} \) |
\(\ p\wedge{r} \) |
\(\ p\wedge{(q\vee{r})} \) |
\(\ (p\wedge{q})\vee{(p\wedge{r})} \) |
| \(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ T \) |
\(\ F\) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
Identity Law
The Identity Law in propositional logic refers to the principle that certain operations, when applied to a statement alongside a specific identity element, yield the original statement. In the context of AND and OR operations, the identity elements are typically "true" and "false" respectively. Applying an AND operation to a statement with "true" yields the original statement, while applying an OR operation to a statement with "false" yields the original statement.
- \( p\vee F\equiv p\)
| \(\ p \) |
\(\ F\) |
\(\ p\vee F \) |
| \(\ T \) |
\(\ F\) |
\(\ T \) |
| \(\ F \) |
\(\ F\) |
\(T \) |
- \( p\vee T\equiv T\)
| \(\ p \) |
\(\ T\) |
\( p\vee T\) |
| \(\ T \) |
\(\ T\) |
\(\ T \) |
| \(\ F \) |
\(\ T\) |
\(T \) |
- \( p\wedge T \equiv p\)
| \(\ p \) |
\(\ T\) |
\(\ p\wedge T \) |
| \(\ T \) |
\(\ T\) |
\(\ T \) |
| \(\ F \) |
\(\ T\) |
\(F\) |
- \( p\wedge F\equiv F\)
| \(\ p \) |
\(\ F\) |
\(\ p\wedge F \) |
| \(\ T \) |
\(\ F\) |
\(\ F \) |
| \(\ F \) |
\(\ F\) |
\(F\) |
Complement Law
The Complement Law, also known as the Law of Double Negation, states that a statement and its negation are mutually exclusive and exhaustive, meaning one of them must be true while the other is false. This law emphasizes the relationship between a statement and its negation, highlighting that they encompass all possibilities within a logical system.
- \( p\vee \sim p\equiv T \)
| \(\ p \) |
\(\ \sim{p} \) |
\(\ p \vee{\sim p} \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
| \(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ T \) |
- \( \sim (\sim p)\equiv p \)
| \(\ p \) |
\(\ \sim{p} \) |
\(\ \sim(\sim p) \) |
| \(\ T \) |
\(\ F \) |
\(\ T \) |
| \(\ F \) |
\(\ T \) |
\(\ F \) |
- \( p\wedge \sim p\equiv F\)
| \(\ p \) |
\(\ \sim{p} \) |
\(\ p \wedge{\sim p} \) |
\(\ F \) |
| \(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ F \) |
| \(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
- \( \sim T \equiv F\) ; \( \sim F \equiv T\)
| \(\ T \) |
\(\ \sim{T} \) |
\(\ F \) |
\(\ \sim{F} \) |
| \(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
| \(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
De-Morgan's Law
De Morgan's Law is a pair of fundamental principles that describe the relationship between negated logical operations. It states that the negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negations of the operands, and vice versa. Similarly, it asserts that the negation of a disjunction is equivalent to the conjunction of the negations of the operands, and vice versa. De Morgan's Law provides a useful tool for transforming complex logical expressions into equivalent, often more manageable forms.
- \( \sim (p\vee q)\equiv \sim p\wedge \sim q \)
| \(\ p \) |
\(\ q \) |
\(\ \sim{p} \) |
\(\ \sim{q} \) |
\(\ p\vee{q} \) |
\(\sim(p\vee{q}) \) |
\(\ \sim{p}\wedge \sim{q} \) |
| \(\ T\) |
\(\ T\) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
| \(\ T\) |
\(\ F\) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
\(\ F\) |
\(\ F\) |
| \(\ F\) |
\(\ T\) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ F\) |
\(\ F\) |
| \(\ F\) |
\(\ F\) |
\(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
- \( \sim (p\wedge q)\equiv \sim p\vee \sim q \)
| \(\ p \) |
\(\ q \) |
\(\ \sim{p} \) |
\(\ \sim{q} \) |
\(\ p\wedge{q} \) |
\(\sim(p\wedge{q}) \) |
\(\ \sim{p}\vee \sim{q} \) |
| \(\ T\) |
\(\ T\) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
| \(\ T\) |
\(\ F\) |
\(\ F \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
| \(\ F\) |
\(\ T\) |
\(\ T \) |
\(\ F \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
| \(\ F\) |
\(\ F\) |
\(\ T \) |
\(\ T \) |
\(\ F \) |
\(\ T \) |
\(\ T \) |
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