Laws and Rules of Algebra

Question 1

Commutative Law for Addition​

Answer 1

A + B = B + A

Question 2

Commutative Law for Multiplication​

Answer 2

A * B = B * A​

Question 3

Associative Law for Addition​

Answer 3

(A + B) + C = A + (B + C)​

Question 4

Distributive Law

Answer 4

A(B + C) = (A * B) + (A * C)

Question 4

Associative Law for Multiplication​

Answer 4

(A * B) * C = A * (B * C)

Question 5

Substitution

Answer 5

If A = 2x, and 2B = A, then B = 2 * (2X)

Question 6

Existential Instantiation

Answer 6

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.

Question 7

Definition of Odd

Answer 7

An integer n is odd if, and only if, n = 2k + 1

Question 8

Definition of Even

Answer 8

An integer n is even if, and only if, n = 2k, where k is any integer

Question 9

Theorem

Answer 9

In mathematics, a theorem refers to a statement that is known to be true because it has been proved.

Question 10

X

Answer 10

Particular but arbitrarily chosen item in domain or set

Question 11

Products and sums of integers

Answer 11

Where w, x, y, and z are integers, the products and sums of integers will be an integer. Thus (xy + wz) is an integer.

Question 12

Counterexamples

Answer 12

Counterexamples are used to disprove universal statements

Question 13

Prime number

Answer 13

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.

Question 14

Direct Proof

Answer 14

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.

Question 15

Rational

Answer 15

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

Question 16

Zero Product Property

Answer 16

The product of two numbers is non-zero if neither number is zero.

Question 17

Rational Integers

Answer 17

Every integer is a rational number of the form x/y

Question 18

Sum/Diff/Prod Even Integers

Answer 18

Sum, product and difference of any two even integers are even

Question 19

Sum/Diff of Odd integers

Answer 19

The sum and difference any two odd integers are even.

Question 20

Product of 2 Odd Integers

Answer 20

The product of two odd integers is odd

Question 21

Sum of Odd and Even Integers

Answer 21

The Sum of an odd and an even integer is odd

Question 22

Diff of Odd and Even Integers

Answer 22

Any odd integer minus any even integer is odd. Any even integer minus any odd integer is odd.

Question 23

Product of Odd and Even Integer

Answer 23

The product of an even integer and an odd integer is even.

Question 24

Divisors of 1

Answer 24

The only divisors of 1 are 1 and -1

Question 25

Positive Divisors of a Positive Integer

Answer 25

For all integers a and b, if a and be are positive and a divides b, then a <= b.

Question 26

A | B [Divides]

Answer 26

A divides B. A|B means that for some integer r, B = A*r

Question 27

Unique Factorization of Integers Theorem

Answer 27

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.

Question 28

Standard Factored Form

Answer 28

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.

Question 29

Quotient Remainder Theorem

Answer 29

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

Question 30

Parity Property

Answer 30

Any given integer is either even or odd

Question 31

Modulus

Answer 31

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

Question 32

Triangle inequality

Answer 32

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|

Question 33

Absolute Value

Answer 33

|x| (the absolute value of x)
x if x >= 0 and
-x if x < 0

Question 34

Break into Cases

Answer 34

A method of proof that functions like a switch statement.

Case 1 (value):
Case 2 (value):

Question 35

Floor

Answer 35

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

Question 36

Ceiling

Answer 36

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

Question 36

Finding Modulus

Answer 36

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

Question 37

Modus Ponens

Answer 37

If P, then Q. We know P, therefore Q

Question 38

Modus Tollens

Answer 38

If P then Q. We know NOT Q, therefore NOT P.

Question 39

Inferences

Answer 39

Given X means P or Q
We know P.
Therefore X.

Question 40

Specialization

Answer 40

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.

Question 41

Elimination

Answer 41

Given P or Q is required. ~Q, then P. ~P then Q.

Question 42

TRansitivity

Answer 42

P, therefore Q.
Q therefore R.
Thus, P therefore R.

Question 43

Division into Cases (Simple)

Answer 43

P therefore Q or R therefore Q.
Therefore if P or Q then R.

Question 44

Converse Error
Fallacy of Affirming the Consequent

Answer 44

INVALID ARGUMENT
If P then Q.
Q, therefore P.

Question 45

Inverse Error
Fallacy of Denying the Antecedent

Answer 45

INVALID ARGUMENT
If P, then Q.
~P, therefore ~Q

Question 46

Sound Argument

Answer 46

Sound if and only if valid and all its premises are true.

Question 47

Contradiction

Answer 47

~P therefore C, where C is a contradiction. Therefor, P.

Question 48

Logical Equivalencies

Answer 48

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

Question 49

Commutative Laws

Answer 49

P || q === q || p
P && q === q && p

Question 50

Associative Laws

Answer 50

(P or Q) or R === P or (Q or R)
(P and Q) and R === P and (Q and R)

Question 51

Distributive Laws

Answer 51

P and (Q or R) === (P and Q) or (P and R)
P or (Q and R) === (P or Q) and (P or R)

Question 52

Identity Laws

Answer 52

P and T === P
P or C === P

Question 53

Negation Laws

Answer 53

P or ~P === T
P and ~P === C

Question 54

Double Negative Law

Answer 54

~(~P) === P

Question 55

Idempotent Laws

Answer 55

P and P === P
P or P === P

Question 56

Universal Bound Laws

Answer 56

P or T === T
P and C === C

Question 57

De Morgan’s Laws

Answer 57

~(P and Q) === ~P or ~Q
~(P or Q) === ~P and ~Q
Flip inner connector, not them both

Question 58

Absorption Laws

Answer 58

P or (P and Q) ==- P
P and (P or Q) === P

Question 59

Negations of T and C

Answer 59

~T === C
~C === T