Laws and Rules of Algebra Flashcards
Learn them.
Commutative Law for Addition
A + B = B + A
Commutative Law for Multiplication
A * B = B * A
Associative Law for Addition
(A + B) + C = A + (B + C)
Distributive Law
A(B + C) = (A * B) + (A * C)
Associative Law for Multiplication
(A * B) * C = A * (B * C)
Substitution
If A = 2x, and 2B = A, then B = 2 * (2X)
Existential Instantiation
If it ix known to exist, it can be given a name (e.g. there exist an integer, let us call it x). Name must be unique to scope.
Definition of Odd
An integer n is odd if, and only if, n = 2k + 1
Definition of Even
An integer n is even if, and only if, n = 2k, where k is any integer
Theorem
In mathematics, a theorem refers to a statement that is known to be true because it has been proved.
X
Particular but arbitrarily chosen item in domain or set
Products and sums of integers
Where w, x, y, and z are integers, the products and sums of integers will be an integer. Thus (xy + wz) is an integer.
Counterexamples
Counterexamples are used to disprove universal statements
Prime number
A number is prime if, and only if:
x > 1
AND
if n = x * y, then x or y = 1 and y or x = n.
Direct Proof
for every x in set D, if P(X), then Q(x)
Suppose that x is a particular but arbitrarily chosen element of D that makes the hypothesis P(x) true, and then show that x makes the conclusion (Q(x)) also true.
Rational
A real number r is rational if, and only if, it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational.
R is rational <=> there exists integers A and B such that R = A/B and b != 0
Zero Product Property
The product of two numbers is non-zero if neither number is zero.
Rational Integers
Every integer is a rational number of the form x/y
Sum/Diff/Prod Even Integers
Sum, product and difference of any two even integers are even
Sum/Diff of Odd integers
The sum and difference any two odd integers are even.
Product of 2 Odd Integers
The product of two odd integers is odd
Sum of Odd and Even Integers
The Sum of an odd and an even integer is odd
Diff of Odd and Even Integers
Any odd integer minus any even integer is odd. Any even integer minus any odd integer is odd.
Product of Odd and Even Integer
The product of an even integer and an odd integer is even.
Divisors of 1
The only divisors of 1 are 1 and -1
Positive Divisors of a Positive Integer
For all integers a and b, if a and be are positive and a divides b, then a <= b.
A | B [Divides]
A divides B. A|B means that for some integer r, B = A*r
Unique Factorization of Integers Theorem
Any integer greater than 1 either is prime or can be written as a product of prime numbers in a way that is unique except for the order in which primes are written.
Standard Factored Form
Given any integer greater than 1, n is an expression of the form n = p1 * p2 *p3 * pk where k is a positive integer and P represents a prime number, and P values are listed in ascending order.
Quotient Remainder Theorem
When any integer n is divided by any positive integer d, result is quotient q and a nonnegative integer remainder r that is smaller than d.
Given any integer n and positive integer d, there exists unique integers q and r such that n = dq +r ann d0 <= r < d
Parity Property
Any given integer is either even or odd
Modulus
In general, according to the quotient-remainder theorem, if an integer n is divided by integer d, the possible remainders are 0, 1, 2, 3 . . .(d-1). Tis implies that n can be written as dq + remainder
Triangle inequality
The absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.
For all real numbers x and y, |x + Y| <= |x| + |Y|
Absolute Value
|x| (the absolute value of x)
x if x >= 0 and
-x if x < 0
Break into Cases
A method of proof that functions like a switch statement.
Case 1 (value):
Case 2 (value):
Floor
Given any real number x, the floor of x [x] is a unique integer n such that n <= x < n+1. Floor of x = n if and only if n <= x < n+1
Ceiling
Given any real number x, the ceiling of x ([x]) is that unique integer n such that n-1 < x <= n. N is the integer that satisfies [x] = n if and only if n -1< x <=n
Finding Modulus
If n is any integer and d is a positive integer, and if q = floor(n/d), and r = n - d &* floor(n/d), then n = dq + r and 0 <= r < d
Modus Ponens
If P, then Q. We know P, therefore Q
Modus Tollens
If P then Q. We know NOT Q, therefore NOT P.
Inferences
Given X means P or Q
We know P.
Therefore X.
Specialization
Given X has properties P and Q, we know therefor that X has property P (desired property).
goal is to foxcus on property of interest.
Elimination
Given P or Q is required. ~Q, then P. ~P then Q.
TRansitivity
P, therefore Q.
Q therefore R.
Thus, P therefore R.
Division into Cases (Simple)
P therefore Q or R therefore Q.
Therefore if P or Q then R.
Converse Error
Fallacy of Affirming the Consequent
INVALID ARGUMENT
If P then Q.
Q, therefore P.
Inverse Error
Fallacy of Denying the Antecedent
INVALID ARGUMENT
If P, then Q.
~P, therefore ~Q
Sound Argument
Sound if and only if valid and all its premises are true.
Contradiction
~P therefore C, where C is a contradiction. Therefor, P.
Logical Equivalencies
Commutative Laws
Associative Laws
Distributive Laws
Identity laws
Negation Laws
Duble Negative Law
Idempotent Laws
Universal Bound Laws
De Morgan’s Laws
Absorption Laws
Negations of T and C
Commutative Laws
P || q === q || p
P && q === q && p
Associative Laws
(P or Q) or R === P or (Q or R)
(P and Q) and R === P and (Q and R)
Distributive Laws
P and (Q or R) === (P and Q) or (P and R)
P or (Q and R) === (P or Q) and (P or R)
Identity Laws
P and T === P
P or C === P
Negation Laws
P or ~P === T
P and ~P === C
Double Negative Law
~(~P) === P
Idempotent Laws
P and P === P
P or P === P
Universal Bound Laws
P or T === T
P and C === C
De Morgan’s Laws
~(P and Q) === ~P or ~Q
~(P or Q) === ~P and ~Q
Flip inner connector, not them both
Absorption Laws
P or (P and Q) ==- P
P and (P or Q) === P
Negations of T and C
~T === C
~C === T
