Algebra of Statement: Some Standard Equivalence Relation

Mathematics Class XI Chapter 1: Mathematical Reasoning Algebra of Statement: Some Standard Equivalence Relation

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Some standard equivalence relation

Idempotent law:

$p \cup p \equiv p$
$p \cap p \equiv p$

Identity law:

$p \cup F \equiv p$
$p \cap F \equiv F$
$p \cup T \equiv T$
$p \cap T \equiv p$

Complement law:

$p \cup \sim p \equiv T$
$p \cap \sim p \equiv F$

Involution law:

$\sim T \equiv F$
$\sim F \equiv T$
$\sim \left( { \sim p} \right) \equiv p$

Commutative law:

$p \cup q \equiv q \cup p$
$p \cap q \equiv q \cap p$

Associative laws:

$\left( {p \cup q} \right) \cup r \equiv p \cup \left( {q \cup r} \right) \equiv p \cup q \cup r$
$\left( {p \cap q} \right) \cap r \equiv p \cap \left( {q \cap r} \right) \equiv p \cap q \cap r$

Distributive law:

$p \cup \left( {q \cap r} \right) \equiv \left( {p \cup q} \right) \cap \left( {p \cup r} \right)$
$p \cap \left( {q \cup r} \right) \equiv \left( {p \cap q} \right) \cup \left( {p \cap r} \right)$

Absorption law:

$p \cup \left( {p \cap q} \right) \equiv p$
$p \cap \left( {p \cup q} \right) \equiv p$

Conditional law:

$p \to q \equiv \sim p \cup q$
$p \leftrightarrow q \equiv \left( { \sim p \cup q} \right) \cap \left( { \sim q \cup p} \right)$

De Morgan’s law

$\sim \left( {p \cup q} \right) \equiv \sim p \cap \sim q$
$\sim \left( {p \cap q} \right) \equiv \sim p \cup \sim q$