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COMP 283 Midterm 1 Study Guide
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Terms in this set (34)
Disjunction (or)
Denoted by "V"
Conjunction (and)
Denoted by "Λ"
Implication (ex. p⊃q)
Defined as ¬p V q (or p → q)
Equivalence (ex. p≡q)
Defined as (p⊃q) Λ (q⊃p)
Exclusive or (ex. p⊕q)
Defined as (p Λ ¬q) V (q Λ ¬p)
Tautology
A formula is one if ALL variations satisfy it. Ex. p V ¬p
Contradiction (or unsatisfiable)
A formula is one if NO valuation satisfies it. Ex.p Λ ¬p
Satisfiable
A formula is one if it has AT LEAST ONE satisfying valuation. Ex. p V q
∈
belongs to; membership
∉
doesn't belong to
{}
empty set or ∅
A⊆B
subset; A is a subset of B if all elements of A are elements of B
AUB
(A union B); is defined by x ∈ AUB if x∈A or x∈B
A∩B
(A intersection B); is defined by x∈A∩B if x∈A and x∈B
A = B
If A⊆B and B⊆A
|S| (size or cardinality of sets)
|#| indicates the number of elements in a set. This can be a nonnegative integer or even infinity.
Cartesian product
If A and B are sets then AxB is {(a, b): a∈A; b∈B}
Ax∅
∅
Permutation
Permutations of n distinct items are the set of all n-tuples that never repeat any item. This is the set of different orders for the n items. *
ORDER MATTERS
*
Combination
A r-combination is a set with r elements chosen from a set of n possible items. *
ORDER DOESN'T MATTER
*
∀: universal quantification
∀ x: P(x) or (x) P(x) means P(x) is true for all x.
→
implies; if .. then
∃: existential quantification
∃ x: P(x) means there is at least one x such that P(x) is true.
Set difference (means A-B)
x∈A \ B iff x∈A Λ x ∉ B
iff
"if and only if"
Symmetric difference (exclusive or)
x∈A ⊕ B iff x∈((A \ B) U (B \ A))
\
set difference; Ex. B \ C is the set of elements in B but not in C
partial function
A relation R⊆AxB is called a partial fraction if for all x in A there is at most one y in B such that xRy.
surjection
A function f: A→B is a surjection(onto) if for each y in B there is an x in A such that f(x)=y.
injection
A function f: A→B is an injection(one-to-one) if for each y in B there is at most one x in A such that f(x) = y.
Bijection
A function is a bijection if it's an injection and a surjection.
big O
f ∈ O(g) if there is a c∈R+ and an N∈N such that for all n>N, f(n) ≤ cg(n)
big omega
f ∈ Ω (g) if there is a c ∈ R+ and an N∈N such that for all n>N, f(n) ≥ cg(n)
big theta
f ∈ θ(g) if f ∈ O(g) and f ∈ Ω (g)
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