In this problem, your task is to list all the constants, free variables and bound variables in each of these formulas in the appropriate columns (if there are any). If there are none, write “NONE” under the appropriate column.
In this problem, your task is to list all the constants, free variables and bound variables in each of these formulas in the appropriate columns (if there are any). If there are none, write “NONE” under the appropriate column.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In this problem, your task is to list all the constants, free variables and bound variables in each of these formulas in the appropriate columns (if there are any). If there are none, write “NONE” under the appropriate column.
(The image with the chart is the question)
(The other image is just formulas if your not familiar with this)
Transcribed Image Text:Formula
Lxa
EXEL
Lax
3xLxa
Lxx
3xLxy
Lxy
3xLax
Laa
3XLXX
(Lzz → Lyx)
vz³(→ Lyz)
Constants
Bound Variables
Free Variables
Transcribed Image Text:We are now in a position do describe quantified relational logic
for one or two place relations only. Remember that to specify a
logic I need to tell you (1) the formal symbols, (2) the
transformation rules, and (3) closure condition.
The formal symbols are:
· a, b, c, . . . , m, n, . . . as constant symbols for terms. A term is
anything in a theory or language that can be given a proper
name or an object that can be identified uniquely.
- x, y, z as symbols for variables, which range over terms.
- Upper case letters P, Q, R, S, T, . . . , A, B, C, D, . . . , M, . . . of
alphabet as symbols for relations (on two place
the
only).
- Truth-functional connectives: V, ^, , , ↔
- Quantifiers: V
- Brackets: ( for left bracket and ) for right bracket.
In order to talk about formulas at a meta-level we use the
symbols F, G and H.
The transformation rules for quantified relational logic are:
1. For any predicate symbol P and for any two place relation
symbol R, given any constants a or b or any variables x or y,
Pa, Px, Rab, Rxy are formulas. In Px and Rxy formulas, x and
y are said to "free" variables because there are no quantifiers to
which they are bound.
2. If F is a formula by Rule 1 and if x is a free variable in F, then
3x F is a formula and Vx F is a formula. Rule 2 is known as
binding any free variable x in F by a quantifier.
A formula F formed by either rule 1 and 2 is called an atomic
formula.
3. If F and G are atomic formulas, then -F, (F v G), (F ^ G),
· G) and (F ↔ G) are complex formulas.
(F
4. If H is a complex formula, then the result of binding any free
variable in H is a formula.
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