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Summary Hunter college Discrete structures study guide
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Summary Hunter college Discrete structures study guide
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Summary Hunter college Discrete structures study guide∀x P(x) - ANSWER "For all x, P(x)"
∃x P(x) - ANSWER "There exists some x, such that P(x)"
x∈A - ANSWER "x is an element of A"
x∉A - ANSWER "x is not an element of A"
p
p→q
_______
∴ q - ANSWER Modus Ponens
-q
p→q
_______
∴-p - ANSWER Modus tollens
∀xP(x)
___________
∴P(c) - ANSWER Universal Instantiation
P(c) for an arbitrary c
___________
∴∀xP(x) - ANSWER Universal Generalization
∃xP(x)
___________
∴P(c) for some element c - ANSWER Existential Instantiation
P(c) for some element c
___________
∴∃xP(x) - ANSWER Existential Instantiation
If a, b, and c are integers, - ANSWER 1) if a/b and a/c, then a/(b+c)
2) if a/b,, then a/bc for all integers c
3) if a/b and b/c, then a/c
If a, b, c are integers such that a/b and a/c - ANSWER then a/(mb+nc)
An integer is called "prime" if - ANSWER the only positive factors of p are 1 and p.
An integer is called "composite" if - ANSWER it is not prime.
, Integers a & b are "relatively prime" - ANSWER if their greatest common divisor is 1.
Even Integer - ANSWER An integer that can be written as 2n, where n is some
integer.
Odd Integer - ANSWER An integer that can be written as 2n+1, where n is some
integer.
Prime Integer - ANSWER A integer n>1 whose only multiples are 1 and itself. For
positive integers r and s,
If n=rs, then r=n and s=1, or r=1 and s=n.
Composite Integer - ANSWER There are positive integers r and s such that
n=rs and
1<r<n
1<s<n
(r and s are both multiples of n and less than n.)
To prove an existential statement... - ANSWER Find an x that works for the
statement.
To disprove a universal statement... - ANSWER Find a counterexample.
Exhaustion - ANSWER Prove a universal statement, test all x in the domain.
Rational Number - ANSWER A number that can be written as a/b, where a and b
are integers and b is not 0.
Divisibility - ANSWER If n and d are integers and d is not 0, then n is divisible by d
iff n=dk, where k is an integer. (d|n means n is divisible by d).
Unique Factorization Theorem - ANSWER Every whole number can be uniquely
factored as a product of primes. This result guarantees that if the prime factors are
ordered from smallest to largest, everyone will get the same result when breaking a
number into a product of prime factors.
Universal Quantifiers - ANSWER "For all," symbol: ∀
Existential Qualifiers - ANSWER "There exists," symbol: ∃
∀x P(x) - ANSWER is read as "For all x, P(x)" or "For every x, P(x)"
∃x P(x) - ANSWER is read as "For some x, P(x)", or as "There is an x such that
P(x)," or "For at least one x, P(x)."
Uniqueness Quantifier - ANSWER "There is a unique x such that P(x)" or "There is
one and only one x such that P(x)," symbol: ∃!
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