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Filters and ideals
and ai Ui = i (U) there is some bi Ui i (A). It follows that the function
b: i # bi is in U A, as required.
We remark that the above proof uses the Axiom of Choice in several different
places. Firstly there is an implicit applicatio
7
Valuations
7.1 Semantics for propositional logic
Following the general method for other formal systems in this book, we must
connect the system for propositional logic of the last chapter with the boolean
algebras of the chapter preceding it, by using b
108
Filters and ideals
Exercise 8.24 (For those who have read Section 5.3 on the algebra of Boole.)
Let R be a boolean ring and B the boolean algebra corresponding to it. Show
that I R is an ideal of R in the sense of ring theory (i.e. is non-empty, is
cl
96
Valuations
that there are 2100 individual valuations to check, and this is an unreasonably
large number to expect any sort of computer to run through in a human lifetime.
And of course n = 1000 would be much much worse.
The other algorithm does not far
90
Valuations
since 2 = cfw_, is an example boolean algebra that can be used in the definition of ! . Thus we have proved the following.
Corollary 7.15 For any set of boolean terms and any other statement ,
! if and only if !2 .
This is hardly obvious,
66
Propositional logic
Formal proof
(a b)
a
a
a
b
( a b)
b
(a b)
(1) Assumption
(2) -Elimination
(3)
(4)
Assumption
Contradiction
(5) RAA
(6) -Elimination
(7) Given, from
(8) -Elimination
(9) Contradiction
(10)
RAA
Example 6.5 Let X = cfw_a, b. Then (a b
58
Boolean algebras
The distributivity laws were already given as axioms of boolean algebras so
there is nothing to prove in the next point.
Since is the least element, it must be the greatest lower bound of itself and
any other a X. Hence a = . The other
82
Valuations
Now consider:
a
b
a b ( a b) (b ( a b) (a (b ( a b)
So v(a (b ( a b) is for all valuations.
Finally,
a
b
c
b (a b) (a b) c)
Thus (a b) c) has value for some valuations and for others. We
leave the reader to check the remaining examples.
No
50
Deductions in posets
equivalent to the full Axiom of Choice. In the rest of this section we shall look
at other examples where these systems of proof can help organise and prove
some results in algebra related to ordered structures; all of these will r
6
Propositional logic
6.1 A system for proof about propositions
We are going to develop a formal system for proofs about boolean algebras,
just as in a previous chapter we developed one for posets. It will also be rich
enough to simulate proofs in the sys
68
Propositional logic
We follow with some instructive examples of incorrect proofs.
Example 6.7 Consider the following erroneous proof the shows b is a consequence of (a b). Of course, no such proof should be possible.
Formal proof
(a b)
b
a
b
b
b
(1)
Gi
106
Filters and ideals
in free(X)/G, so this valuation does not send BT(X) to in free(X)/G.
Therefore 1 G is consistent by soundness.
Definition 8.15 Let F be a filter in a boolean algebra B. We say that F is prime
if it is proper and whenever a, b B with
126
First-order logic
Formal proof
x (x)
x (x)
Let a satisfy (a)
(a)
x (x)
(1)
Given
(2)
Assumption
(3)
(4) -Elimination
(5)
(6) -Elimination
(7)
RAA
Exercise 9.15 Prove that x (x) x (x). (Use -Elimination and Elimination.)
The last four examples and ex
92
Valuations
where each j is a conjunction j, 1 j, 2 . . . j, l and each j, i is a propositional letter or the negation of a propositional letter.
Exercise 7.22 By considering the rows of the truth table for which vi = , or
otherwise, show that is also l
8
Filters and ideals
8.1 Algebraic theory of boolean algebras
In this chapter we start to explore the theory of boolean algebras as an algebraic
theory in its own right, in a way analogous to ring theory, say. We will see
many applications of the Complete
84
Valuations
Example 7.9 Again, with proposition letters p, q we have
(q r), (p r) !2 p ( q r).
The truth table is
p
q
r
(q r) (p r)
p ( q r)
Here, we see that the valuation v(p) = , v(q) = , v(r) = makes (q r)
and p r both equal to but makes the conclus
102
Filters and ideals
Definition 8.6 A filter of B is a non-empty subset F B such that for all
x, y B: (a) x ! y F implies x F; and (b) x, y F implies x y F. It is
proper if it is not equal to the whole of B.
As mentioned, filters and ideals are dual con
76
Propositional logic
If and ( ) have been deduced from then may be deduced
from in one further step.
If and ( ) have been deduced from then may be deduced
from in one further step.
If and ( ) have been deduced from then may be deduced
from in one fur
52
Deductions in posets
Proposition 4.32 Suppose G is torsion-free and abelian, x1 , . . ., xk G are all
non-zero, and
cfw_0 x1 , 0 x2 , . . ., 0 xk 0 y
for some y G. Then there are ni N, not all zero, and m N such that
k
my = i=1 ni xi .
Proof By induct
104
Filters and ideals
In the sequel, I will tend to concentrate on filters rather than ideals since (being
an optimist) I prefer to focus on true statements rather than false ones, but as
we have seen these ideas are interchangeable. (Is this boolean alg
54
Deductions in posets
The following exercise gives a family of examples that are right-orderable.
In attempting it, it may be helpful to know that any set such as has a wellorder, i.e. a linear order in which every non-empty subset has a least element.
132
First-order logic
As is often the case with formal proofs, there is rarely a single proof for a
particular statement, and in this case a quite different alternative can be given.
Formal proof
a x ( (x) (a)
Let a be arbitrary
(a)
Let x be arbitrary
(
9
First-order logic
9.1 First-order languages
Propositional logic is the logic of statements that can be true or false, or take
some value in a boolean algebra. The logic of most mathematical arguments
involves more than just this: it involves mathematica
72
Propositional logic
Formal proof
( )
.
(1) Given
(2) -Elimination
(3) -Elimination
(4)
(5)
Example 6.12 To prove a statement from ( ) and other given statements , deduce both of and first, as shown in the following argument,
and then prove from , and .
138
First-order logic
Exercise 10.17 for a hint.) In contrast, some cases can be done: we can describe all simple groups of order 168 (or any other finite order) by a first-order
sentence, for example.
Second-order logic allows for these subset-quantifier
128
First-order logic
Formal proof
x y (P(x) P(y) (x = y)
x (P(x) R(x)
Let x be arbitrary
R(x)
Let a satisfy P(a) R(a)
P(a)
R(a)
P(x)
(P(x) P(a)
y (P(a) P(y) (a = y)
(P(a) P(x) (a = x)
(a = x)
R(x)
P(x)
P(x)
R(x) P(x)
x (R(x) P(x)
(1)
(2)
Given
Given
120
First-order logic
have accidently omitted some brackets.) The idea is that the scope of the quantifier is the part of the formula for which the variables meaning is modified by
the quantifier.
Definition 9.6 A formula is closed or is a sentence if eve
118
First-order logic
We start by defining terms, the expressions representing a mathematical object, such as a number. One problem is that we want to use our non-logical
function symbols to form terms, but some, like 1 for reciprocal, take only one
argum
86
Valuations
of boolean algebras, i.e. any whose valuation is at least as true as that of
however these valuations are chosen, actually has a formal derivation from
assumptions in the formal system. This is hardly an obvious assertion since
the rules fo