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Sets and Logic
Set vocabulary, Venn diagrams and basic logic statements.
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- SetTermA collection of distinct objects
- Element (member)TermAn object that is in a set
- Empty setTermThe set with no elements, written ∅ or { }
- SubsetTermA set whose elements are all in another set
- Proper subsetTermA subset that is not the whole set
- Union (A ∪ B)TermElements in A or B or both
- Intersection (A ∩ B)TermElements in both A and B
- Complement (A′)TermElements in the universal set that are not in A
- Universal setTermThe set of everything under consideration
- Disjoint setsTermSets with no elements in common
- CardinalityTermThe number of elements in a set, written |A|
- Roster formTermListing the elements: {1, 2, 3}
- Set-builder notationTerm{x | x > 2} means all x such that x is greater than 2
- Venn diagramTermOverlapping circles showing how sets relate
- Cardinality of a unionTerm|A ∪ B| = |A| + |B| − |A ∩ B|
- If A = {1,2,3} and B = {3,4}, A ∪ BTerm{1, 2, 3, 4}
- If A = {1,2,3} and B = {3,4}, A ∩ BTerm{3}
- Number of subsets of a set with n elementsTerm2ⁿ
- Number of subsets of {a, b, c}Term8
- Is ∅ a subset of every set?TermYes
- Natural numbers symbolTermℕ
- Integers symbolTermℤ
- Rational numbers symbolTermℚ
- Real numbers symbolTermℝ
- Statement (proposition)TermA sentence that is either true or false
- NegationTermThe statement with the opposite truth value (not P)
- ConjunctionTermP and Q; true only when both are true
- DisjunctionTermP or Q; true when at least one is true
- ConditionalTermIf P then Q; false only when P is true and Q is false
- HypothesisTermThe 'if' part of a conditional
- ConclusionTermThe 'then' part of a conditional
- ConverseTermSwap the parts: if Q then P
- InverseTermNegate both parts: if not P then not Q
- ContrapositiveTermSwap and negate: if not Q then not P; same truth as the original
- BiconditionalTermP if and only if Q; true when both have the same truth value
- Truth tableTermA table listing the truth value of a statement for every case
- TautologyTermA statement that is always true
- ContradictionTermA statement that is always false
- De Morgan's law 1Termnot (P and Q) is the same as (not P) or (not Q)
- De Morgan's law 2Termnot (P or Q) is the same as (not P) and (not Q)
- CounterexampleTermAn example showing a statement is false
- Direct proofTermProving a statement by a chain of logical steps from facts
- Proof by contradictionTermAssume the opposite and show it leads to something impossible
- Proof by inductionTermShow it for a first case, then show each case leads to the next
- Inductive reasoningTermReaching a general rule from patterns
- Deductive reasoningTermReaching a conclusion that must follow from given facts
- AxiomTermA basic statement accepted without proof
- LemmaTermA small proven result used to prove a bigger one
- CorollaryTermA result that follows easily from a theorem
- Q.E.D.TermLatin 'quod erat demonstrandum': which was to be shown
- Necessary conditionTermSomething that must be true for another thing to be true
- Sufficient conditionTermSomething that, if true, guarantees another thing
- Quantifier ∀TermFor all
- Quantifier ∃TermThere exists
- Boolean valueTermTrue or false, often written 1 or 0
- AND gate outputTerm1 only if both inputs are 1
- OR gate outputTerm1 if at least one input is 1
- NOT gate outputTermThe opposite of the input
- Binary number systemTermBase 2, using only digits 0 and 1
- Binary 101 in decimalTerm5
- Decimal 10 in binaryTerm1010
- Binary 1111 in decimalTerm15
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