Math 2.0

Learn Year 1

Logic and sets

145 ideas in teaching order. Each idea comes after the ideas that it builds on. 144 of them have a page so far; the others are on the list in grey.

Start with Proposition →

  1. Proposition concept
  2. Negation definition
  3. Conjunction definition
  4. Disjunction definition
  5. Implication definition
  6. Biconditional definition
  7. Truth table concept
  8. Propositional logic concept
  9. Tautology definition
  10. Logical equivalence definition
  11. Law of excluded middle statement
  12. Double negation law statement
  13. De Morgan's laws statement
  14. Converse definition
  15. Contrapositive statement
  16. Necessary and sufficient conditions concept
  17. Modus ponens concept
  18. Predicate definition
  19. Predicate logic concept
  20. Universal quantifier definition
  21. Existential quantifier definition
  22. Uniqueness quantifier definition
  23. Negation of a universal statement statement
  24. Negation of an existential statement statement
  25. Order of quantifiers concept
  26. Mathematical proof concept
  27. Direct proof concept
  28. Proof by contrapositive concept
  29. Proof by contradiction concept
  30. Proof by cases concept
  31. Constructive proof concept
  32. Counterexample concept
  33. Vacuous truth concept
  34. Set definition
  35. Element definition
  36. Set-builder notation concept
  37. Empty set object
  38. Subset definition
  39. Axiom of extensionality statement
  40. Universal set definition
  41. Union definition
  42. Intersection definition
  43. Complement definition
  44. Set difference definition
  45. Symmetric difference definition
  46. Disjoint sets definition
  47. Venn diagram concept
  48. Distributive laws for sets statement
  49. De Morgan's laws for sets statement
  50. Family of sets definition
  51. Union of a family of sets definition
  52. Intersection of a family of sets definition
  53. Power set definition
  54. Ordered pair definition
  55. Cartesian product definition
  56. Russell's paradox statement
  57. Zermelo–Fraenkel set theory concept
  58. Binary relation definition
  59. Reflexive relation definition
  60. Symmetric relation definition
  61. Antisymmetric relation definition
  62. Transitive relation definition
  63. Composition of relations definition
  64. Converse relation definition
  65. Transitive closure definition
  66. Equivalence relation definition
  67. Equivalence class definition
  68. Quotient set definition
  69. Partition of a set definition
  70. Equivalence classes form a partition statement
  71. Correspondence between equivalence relations and partitions statement
  72. Equivalence relation induced by a function definition
  73. Function definition
  74. Domain definition
  75. Codomain definition
  76. Graph of a function definition
  77. Identity function definition
  78. Range of a function definition
  79. Image of a set definition
  80. Preimage of a set definition
  81. Image of an intersection statement
  82. Restriction of a function definition
  83. Indicator function definition
  84. Function composition definition
  85. Associativity of composition statement
  86. Injective function definition
  87. Surjective function definition
  88. Bijective function definition
  89. Composition of injections is injective statement
  90. Composition of surjections is surjective statement
  91. Left inverse definition
  92. Right inverse definition
  93. Injective iff it has a left inverse statement
  94. Surjective iff it has a right inverse statement
  95. Bijective iff it has a two-sided inverse statement
  96. Inverse function definition
  97. Preorder definition
  98. Partial order definition
  99. Total order definition
  100. Hasse diagram concept
  101. Lexicographic order definition
  102. Maximal element definition
  103. Minimal element definition
  104. Greatest element definition
  105. Least element definition
  106. upper bound definition
  107. Chain definition
  108. Antichain definition
  109. Monotone function definition
  110. Order isomorphism definition
  111. Well-founded relation definition
  112. Well-order definition
  113. Transfinite induction statement
  114. Ordinal number definition
  115. Equinumerous sets definition
  116. Cardinality definition
  117. Cardinal number definition
  118. Finite set definition
  119. Infinite set definition
  120. Every infinite set has a countably infinite subset statement
  121. Hilbert's hotel example
  122. Countable set definition
  123. Countably infinite set definition
  124. Cantor pairing function example
  125. Product of countable sets is countable statement
  126. Countable union of countable sets is countable statement
  127. The integers are countable statement
  128. The rationals are countable statement
  129. Uncountable set definition
  130. Cantor's diagonal argument concept
  131. The real numbers are uncountable statement
  132. Cantor's theorem statement
  133. Cardinality of the power set statement
  134. Order of cardinals definition
  135. Schröder–Bernstein theorem statement
  136. Comparability of cardinals statement
  137. Aleph-null object
  138. Cardinality of the continuum object
  139. The reals have the cardinality of the continuum statement
  140. Continuum hypothesis statement
  141. Axiom of choice statement
  142. Choice function definition
  143. Zorn's lemma statement
  144. Hausdorff maximal principle statement
  145. Well-ordering theorem statement