Boolean
see math notation
definition
either true or false
BB == {{ I, O }}
properties
BB < UU, see universal ‹set
booleans can be manipulated through boolean algebra
Operators
operator precedence
see math notation for infix notation › precedence in my math notation
in conventional math notation: \(' \cdot + \oplus\)
Conjunction
aka AND, "min"
notation in my math notation /\ and __
notation in conventional math notation \(\cdot\) or \(\cap\) (may be omitted)
representation truth table
A |
B |
A /\ B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Disjunction
aka OR, "max"
notation in my math notation \/ and ^^
notation in conventional math notation \(+\) or \(\cup\)
representation truth table
A |
B |
A \/ B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Negation
aka NOT
notation in my math notation >< and +
notation in conventional math notation \(a'\) or \(\bar a\) or \(a^c\)
representation truth table
A |
+A |
|---|---|
| 0 | 1 |
| 1 | 0 |
properties
Symmetric Difference
aka exclusive or, XOR, disjunctive union
notation in my math notation >< and +
notation in conventional math notation \(\oplus\) or \(\vartriangle\)
representation truth table
A |
B |
A + B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Implication
aka conditional statement, logical consequence
definition if x = I, then the conditional statement x < y is said to be vacuously true
definition +x > +y is the contrapositive of the statement x < y. they are logically equivalent
definition x > y is the converse of the statement x < y. they are not logically equivalent
definition +x < +y is the inverse of the statement x < y. they are not logically equivalent
note even though the converse and inverse of a conditional statement are not logically equivalent, the converse of a statement is logically equivalent to the inverse of that statement
definition
let S < R < Q (square implies rectangle implies quadrilateral). then,
S is said to be a sufficient condition for R---knowing that S is true allows the deduction that R is true
Q is said to be a necessary condition for R---for R to be true, Q must be true
alternatively, let A < B. then,
A is said to be a sufficient condition for B
B is said to be a necessary condition for A
notation in my math notation < and -| and > and |-
notation in conventional math notation \(\to\) and \(\implies\)
representation truth table
A |
B |
A < B |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
properties
boolean › implication is a partial order
Equivalence
aka XNOR, double implication, equivalence, biconditional statement, bidirectional implication
notatio in my math notation == and =
notation in conventional math notation \(\Leftrightarrow\) or \(\odot\) or \(=\)
representation truth table
A |
B |
A == B |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
properties
boolean › equivalence is a equivalence ‹relation
Difference
notation in my math notation /\ +
notation in conventional math notation \(\backslash\)
representation truth table
A |
B |
A /\ +B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
identities
properties
identities below hold for all BB {x /\ y}
identity x \/ O == x
identity x /\ I == x
excluded middle x \/ +x == I
excluded middle x /\ +x == O
double negation +I == O
double negation +O == I
idempotence x \/ x == x
idempotence x /\ x == x
domination x \/ I == I
domination x /\ O == O
involution ++x == x
properties
properties
properties below hold for all BB {x /\ y /\ z}
commutativity x \/ y == y \/ x
commutativity x /\ y == y /\ x
associativity x \/ y^^z == x^^y \/ z
associativity x /\ y__z == x__y /\ z
distributivity x /\ y^^z == x__y \/ x__z
distributivity x \/ y__z == x^^y /\ x^^z
De Morgan's +x^^y == +x /\ +y
De Morgan's +x__y == +x \/ +y
absorption x \/ +x__y == x
absorption x /\ +x^^y == x
xor x >< y == x__+y \/ y__+x --- me
xor x >< y == >< x = y --- me
implication x < y == +x \/ y
implication x > y == x \/ +y
biconditional x = y == +x /\ +y \/ x __ y
biconditional x = y == x -| y /\ x |- y
and-implies x__y -| z == x -| z \/ y -| z --- me
or-implies x^^y -| z == x -| z /\ y -| z --- me
implies-or x -| y^^z == x -| y \/ x -| z --- me
implies-and x -| y__z == x -| y /\ x -| z --- me
portation x < y -| z == x__y -| z --- https://www.cs.toronto.edu/~hehner/BAUA.pdf
note many boolean laws based on boolean › implications become painfully obvious when seen from the perspective of type theory by the Curry--Howard correspondence. for instance, the boolean portation law corresponds to currying in type theory