Formal Logic Laws

Question 1

Commutative Law

Answer 1

a ^ b = b ^ a

Question 2

Associative Law

Answer 2

a ^ (b ^ c) = (a ^ b) ^ c

Question 3

Distributive Law

Answer 3

a ^ (b V c) c) = (a ^ b) V (a ^ c)

Question 4

Double Negation Law

Answer 4

~~a = a

Question 5

Identity Law

Answer 5

a ^ True = a | a V False = a

Question 6

Negation Law

Answer 6

a ^ ~a = False a V ~a = True

Question 7

Universal Bound Law

Answer 7

a ^ False = False | a V True = True

Question 8

Negation of Universal Law

Answer 8

~True = False | ~False = True

Question 9

Idempotent Law

Answer 9

a ^ a = a | a V a = a

Question 10

De Morgan’s Law

Answer 10

~(p ^ q) = ~p V ~q

Question 11

Absorption Law

Answer 11

a ^ (a V b) = a | a V (a ^ b) = a

Question 12

Conditional Identities

Answer 12

p -> q = ~p V q | p q = q -> q ^ p -> q

Question 13

Commutative Axiom

Answer 13

a * b = b * a

Question 14

Associativity Axiom

Answer 14

a * b * c = (a * b) * c

Question 15

Distributivity Axiom

Answer 15

a * (b + c) = a * b + a * c

Question 16

Identity Elements Axiom

Answer 16

0 + x = x | 1 * x = x

Question 17

Additive Inverse Axiom

Answer 17

x + i = y then x = y + -i

Question 18

Trichotomy of and = Axiom

Answer 18

∀(x, y) [(x > y) v (x < y) v (x = y)]

Question 19

Transitivity of < and > Axiom

Answer 19

∀(x, y) [(x > y ^ y > z) -> (x > z)]

Question 20

Addition of integers preserve inequality Axiom

Answer 20

∀(x, y, z) [(x > y) -> (x + a > y + a)]

Question 21

Multiplication of integers preserve inequality Axiom

Answer 21

∀(x, y, z) [(x > y) -> (x * a > y * a)]

Question 22

Reflectivity Axiom

Answer 22

a = a

Question 23

Transitivity Axiom

Answer 23

a = b, b = c then a = c

Question 24

Symmetry Axiom

Answer 24

a = b iff b = a

Question 25

Axioms of Closure

Answer 25

The sum, difference, and product of any two integers is an integer

Question 26

Integer Axiom

Answer 26

No integer between 0 and 1