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Truth tables

And, or, not, implies, and tautologies.

A truth table lists every combination of truth values for the variables and the value of the whole statement in each case. A statement true in every row is a tautology; one false in every row is a contradiction. Try p implies q and not p or q and compare.

Worked example · truth table of p and q

Discrete-Math · Propositional-Logic

Truth table of p and q

Step by step

  1. p \wedge q

    2 variable(s) → 4 rows. Fill in every combination.

  2. \

    Contingent: true for some inputs, false for others.

Reveal the answer
p \wedge q

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How to: Truth tables

  1. 2 variable(s) → 4 rows. Fill in every combination.
  2. Contingent: true for some inputs, false for others.

Questions people ask

What makes mathematics "discrete"?

It deals with separate, countable objects (integers, graphs, statements) rather than continuous quantities. No limits, no infinitesimals; instead induction, counting and logic.

How does a proof by induction work?

Show the statement for the first case, then show that whenever it holds for n it holds for n + 1. Like dominoes: the first falls, and each knocks over the next.

What order should I take these lessons in?

Logic and proof first, because every later lesson proves things. Then induction, sets, relations and functions, counting, recurrences, graphs, Boolean algebra and automata, and finally algorithms and complexity, which use nearly everything before them.

Is discrete mathematics the same as the maths of computer science?

Largely. Data structures are graphs and trees, correctness arguments are induction, running times are recurrences and sums, circuits are Boolean algebra, and what a computer can do efficiently is the subject of the last lessons. It is also ordinary mathematics in its own right.

Do I need calculus for this course?

No. School algebra is enough. A little familiarity with limits helps when comparing growth rates, and the lesson on asymptotic analysis explains the one limit fact it uses.

More in Discrete Math & Logic

Sums and inductionProof by inductionAlgorithms and growth of functionsPredicate logic and quantifiersDirect proof, contrapositive and contradictionStrong induction and the well-ordering principleRelations: equivalence relations and partial ordersFunctions and cardinalityCounting: rules, choices and bijectionsThe pigeonhole principleSolving linear recurrence relationsGraphs: degrees, paths and connectivityTrees and spanning treesEuler and Hamilton paths