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

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