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MH HSC 12 Maths

Mathematical Logic

Each concept is a short lesson, then questions until you get three right.

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11 concepts, a short lesson each

  • Statements0/3
  • Equivalence0/6
  • Switching circuits0/2

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Statements and truth values

A statement is a sentence that is either true or false, not both; questions, commands and open sentences are not

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Statements0 of 3

  1. Statements and truth valuesA statement is a sentence that is either true or false, not both; questions, commands and open sentences are notStart
  2. Logical connectives∼,∧,∨,→,↔\sim, \wedge, \vee, \to, \leftrightarrow: write a sentence in symbols and find its truth value; p→qp \to q is false only when pp is T and qq is FStart
  3. Truth tablesnn simple statements give 2n2^n rows; build the table column by column, inside brackets firstStart

Equivalence0 of 6

  1. Tautology, contradiction, contingencyAll T in the last column: tautology; all F: contradiction; a mix: contingencyStart
  2. Logical equivalence and lawsEqual last columns mean ≡\equiv; p→q≡∼p∨qp \to q \equiv \sim p \vee q, De Morgan's, distributive and absorption lawsStart
  3. Converse, inverse, contrapositiveFor p→qp \to q: converse q→pq \to p, inverse ∼p→∼q\sim p \to \sim q, contrapositive ∼q→∼p\sim q \to \sim p, which alone is equivalentStart
  4. Negating compound statements∼(p∧q)≡∼p∨∼q\sim(p \wedge q) \equiv \sim p \vee \sim q; ∼(p→q)≡p∧∼q\sim(p \to q) \equiv p \wedge \sim q; ∼(p↔q)≡(p∧∼q)∨(q∧∼p)\sim(p \leftrightarrow q) \equiv (p \wedge \sim q) \vee (q \wedge \sim p)Start
  5. Quantifiers and their negation∀\forall (for all) and ∃\exists (there exists); the negation swaps them and negates the conditionStart
  6. DualityThe dual swaps ∧\wedge and ∨\vee, and tt and cc; ∼\sim stays as it isStart

Switching circuits0 of 2

  1. Switching circuitsSeries is ∧\wedge, parallel is ∨\vee, a switch always opposite to SS is ∼S\sim S; write the statement of a circuit and draw the circuit of a statementStart
  2. Simplifying circuitsSimplify the circuit's statement with the laws of logic, then draw the simpler equivalent circuitStart

Practise Mathematical Logic

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions