Math 2.0

definition

Logical equivalence

Definition

Let φ\varphi and ψ\psi be formulas of propositional logic. The formulas φ\varphi and ψ\psi are logically equivalent when they have the same truth value for each assignment of truth values to their variables. This is written φ≡ψ\varphi \equiv \psi.

Remarks

The formulas φ\varphi and ψ\psi are logically equivalent exactly when φ↔ψ\varphi \leftrightarrow \psi is a tautology. For example, p→q≡¬p∨qp \to q \equiv \neg p \lor q. De Morgan's laws give two more examples.

Explanations

L1 human

Logically equivalent formulas say the same thing in different words. You can check an equivalence with a truth table: the two columns must agree in every row. Then you can replace one formula by the other in a proof.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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What this needs

To read this node, read these 10 nodes in this order. The foundations come first, and the list ends with this node.

Proposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinition
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Proposition concept
  2. Conjunction definition
  3. Disjunction definition
  4. Implication definition
  5. Negation definition
  6. Biconditional definition
  7. Truth table concept
  8. Propositional logic concept
  9. Tautology definition
  10. Logical equivalence definition

What depends on this

12 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Logical_equivalence, lead section

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