{
 "source": "https://peabody.kids/flashcards/math-sets-logic",
 "id": "math-sets-logic",
 "title": "Sets and Logic",
 "category": "math",
 "subject": "Math",
 "levels": [
  "6-8",
  "9-12",
  "college"
 ],
 "description": "Set vocabulary, Venn diagrams and basic logic statements.",
 "tags": [
  "sets",
  "logic",
  "venn diagrams"
 ],
 "frontLabel": "Term",
 "backLabel": "Meaning",
 "cards": [
  [
   "Set",
   "A collection of distinct objects"
  ],
  [
   "Element (member)",
   "An object that is in a set"
  ],
  [
   "Empty set",
   "The set with no elements, written ∅ or { }"
  ],
  [
   "Subset",
   "A set whose elements are all in another set"
  ],
  [
   "Proper subset",
   "A subset that is not the whole set"
  ],
  [
   "Union (A ∪ B)",
   "Elements in A or B or both"
  ],
  [
   "Intersection (A ∩ B)",
   "Elements in both A and B"
  ],
  [
   "Complement (A′)",
   "Elements in the universal set that are not in A"
  ],
  [
   "Universal set",
   "The set of everything under consideration"
  ],
  [
   "Disjoint sets",
   "Sets with no elements in common"
  ],
  [
   "Cardinality",
   "The number of elements in a set, written |A|"
  ],
  [
   "Roster form",
   "Listing the elements: {1, 2, 3}"
  ],
  [
   "Set-builder notation",
   "{x | x > 2} means all x such that x is greater than 2"
  ],
  [
   "Venn diagram",
   "Overlapping circles showing how sets relate"
  ],
  [
   "Cardinality of a union",
   "|A ∪ B| = |A| + |B| − |A ∩ B|"
  ],
  [
   "If A = {1,2,3} and B = {3,4}, A ∪ B",
   "{1, 2, 3, 4}"
  ],
  [
   "If A = {1,2,3} and B = {3,4}, A ∩ B",
   "{3}"
  ],
  [
   "Number of subsets of a set with n elements",
   "2ⁿ"
  ],
  [
   "Number of subsets of {a, b, c}",
   "8"
  ],
  [
   "Is ∅ a subset of every set?",
   "Yes"
  ],
  [
   "Natural numbers symbol",
   "ℕ"
  ],
  [
   "Integers symbol",
   "ℤ"
  ],
  [
   "Rational numbers symbol",
   "ℚ"
  ],
  [
   "Real numbers symbol",
   "ℝ"
  ],
  [
   "Statement (proposition)",
   "A sentence that is either true or false"
  ],
  [
   "Negation",
   "The statement with the opposite truth value (not P)"
  ],
  [
   "Conjunction",
   "P and Q; true only when both are true"
  ],
  [
   "Disjunction",
   "P or Q; true when at least one is true"
  ],
  [
   "Conditional",
   "If P then Q; false only when P is true and Q is false"
  ],
  [
   "Hypothesis",
   "The 'if' part of a conditional"
  ],
  [
   "Conclusion",
   "The 'then' part of a conditional"
  ],
  [
   "Converse",
   "Swap the parts: if Q then P"
  ],
  [
   "Inverse",
   "Negate both parts: if not P then not Q"
  ],
  [
   "Contrapositive",
   "Swap and negate: if not Q then not P; same truth as the original"
  ],
  [
   "Biconditional",
   "P if and only if Q; true when both have the same truth value"
  ],
  [
   "Truth table",
   "A table listing the truth value of a statement for every case"
  ],
  [
   "Tautology",
   "A statement that is always true"
  ],
  [
   "Contradiction",
   "A statement that is always false"
  ],
  [
   "De Morgan's law 1",
   "not (P and Q) is the same as (not P) or (not Q)"
  ],
  [
   "De Morgan's law 2",
   "not (P or Q) is the same as (not P) and (not Q)"
  ],
  [
   "Counterexample",
   "An example showing a statement is false"
  ],
  [
   "Direct proof",
   "Proving a statement by a chain of logical steps from facts"
  ],
  [
   "Proof by contradiction",
   "Assume the opposite and show it leads to something impossible"
  ],
  [
   "Proof by induction",
   "Show it for a first case, then show each case leads to the next"
  ],
  [
   "Inductive reasoning",
   "Reaching a general rule from patterns"
  ],
  [
   "Deductive reasoning",
   "Reaching a conclusion that must follow from given facts"
  ],
  [
   "Axiom",
   "A basic statement accepted without proof"
  ],
  [
   "Lemma",
   "A small proven result used to prove a bigger one"
  ],
  [
   "Corollary",
   "A result that follows easily from a theorem"
  ],
  [
   "Q.E.D.",
   "Latin 'quod erat demonstrandum': which was to be shown"
  ],
  [
   "Necessary condition",
   "Something that must be true for another thing to be true"
  ],
  [
   "Sufficient condition",
   "Something that, if true, guarantees another thing"
  ],
  [
   "Quantifier ∀",
   "For all"
  ],
  [
   "Quantifier ∃",
   "There exists"
  ],
  [
   "Boolean value",
   "True or false, often written 1 or 0"
  ],
  [
   "AND gate output",
   "1 only if both inputs are 1"
  ],
  [
   "OR gate output",
   "1 if at least one input is 1"
  ],
  [
   "NOT gate output",
   "The opposite of the input"
  ],
  [
   "Binary number system",
   "Base 2, using only digits 0 and 1"
  ],
  [
   "Binary 101 in decimal",
   "5"
  ],
  [
   "Decimal 10 in binary",
   "1010"
  ],
  [
   "Binary 1111 in decimal",
   "15"
  ]
 ]
}