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MATH 0050 assessment

创建日期：2024-05-27 15:03:49

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标签： MATH 0050

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MATH 0050 final assessment

1. (a) The set of formulae in the first order predicate language (is denoted by L and) is defined inductively
as follows:
· If P is an n-ary predicate symbol and x1, · · · xn are variable symbols, then Px1 · · ·xn is a
formula.
· If α is a formula, then so is ¬α.
· If α, β are formulae, then so is ⇒ αβ.
· If x is a variable symbol and α is a formula, then ∀xα is a formula.
Let us now consider the given strings:
◦ The string ⇒ ¬Qxy ⇒ ∀x⇒ PxPy is not a formula.
If the string ⇒ ¬Qxy ⇒ ∀x ⇒ PxPy were a formula, then ¬Qxy ⇒ ∀x ⇒ PxPy would
have to consist of two formulae. We note that Qxy is a formula (since Q is a binary, or 2-ary,
predicate symbol), and, hence, ¬Qxy is a formula; so, if the string ¬Qxy ⇒ ∀x ⇒ PxPy
were to consist of two formulae, the string ⇒ ∀x⇒ PxPy would be a formula. For this, latter
string, we note that Px and Py are formulae (since P is a unary, or 1-ary, predicate symbol),
and, hence, that ⇒ PxPy is a formula, and, in turn, that ∀x ⇒ PxPy is a formula. As such,
the string ⇒ ∀x ⇒ PxPy consists of a ⇒ symbol followed, on the right, by a single formula,
namely the formula ∀x⇒ PxPy. None of the defining rules of formulae allows us to construct
a formula that consists of a ⇒ symbol, followed, on the right, by a single formula. Therefore,
the string ⇒ ∀x ⇒ PxPy cannot be a formula, and, hence, as might be indicated above, the
given string, ⇒ ¬Qxy ⇒ ∀x⇒ PxPy, cannot be a formula.
◦ The string ⇒ ∀x⇒ Px⇒ ¬Qxy∀yPyQyx is a formula.
We may use the definition of formula, given above, to construct the string as a formula. We
may obtain Px and Py, as well as Qxy and Qyx, as formulae (since P is a unary, or 1-ary,
predicate symbol, and Q is a binary, or 2-ary, predicate symbol). Since Py is a formula, ∀yPy
is a formula, while, since Qxy is a formula, ¬Qxy is a formula. It follows that, since ∀yPy
and ¬Qxy are formulae, ⇒ ¬Qxy∀yPy is (also) a formula. It follows, in turn, that, since Px
and ⇒ ¬Qxy∀yPy are formulae, ⇒ Px ⇒ ¬Qxy∀yPy is (also) a formula, and, hence, that
∀x ⇒ Px ⇒ ¬Qxy∀yPy is a formula. Finally, since ∀x ⇒ Px ⇒ ¬Qxy∀yPy and Qyx are
formulae, the given string, ⇒ ∀x⇒ Px⇒ ¬Qxy∀yPyQyx, is (also) a formula.
1
(b) Let us transform the given formulae into formulae in L, as required. We start with:
· ((Px) ∨ (¬ (Py)))⇒ Qxy
· (((∀y) (Py)) ∧ (¬ (Px)))⇒ ((∃x) (Qyx))
Note that we cannot use the symbols ‘∃’, ‘∨’, ‘∧’, ‘⇔’, or brackets in L, and we must write strings
of the form ‘α⇒ β’ as ‘⇒ αβ’.
Let us gradually transform these formulae.
First, let us avoid the use of the ‘∨’ symbol above, by using the general convention that states that
‘α ∨ β’ corresponds to ‘(¬α)⇒ β’:
· ((¬ (Px))⇒ (¬ (Py)))⇒ Qxy
· (((∀y) (Py)) ∧ (¬ (Px)))⇒ ((∃x) (Qyx))
Similarly, let us avoid the use of the ‘∧’ symbol above, by using the general convention that states
that ‘α ∧ β’ corresponds to ‘¬(α⇒ (¬β))’:
· ((¬ (Px))⇒ (¬ (Py)))⇒ Qxy
· (¬ (((∀y) (Py))⇒ (¬ (¬ (Px)))))⇒ ((∃x) (Qyx))
Next, let us avoid the use of the ‘∃’ symbol above, by using the general convention that states that
‘(∃x)(α)’ corresponds to ‘¬((∀x)(¬α))’:
· ((¬ (Px))⇒ (¬ (Py)))⇒ Qxy
· (¬ (((∀y) (Py))⇒ (¬ (¬ (Px)))))⇒ (¬ ((∀x) (¬ (Qyx))))
Let us next rewrite strings of the form ‘α ⇒ β’ as ‘⇒ αβ’, starting with the innermost (“most
bracketed”) ‘⇒’ symbols:
· (⇒ (¬ (Px)) (¬ (Py)))⇒ Qxy
· (¬ (⇒ ((∀y) (Py)) (¬ (¬ (Px)))))⇒ (¬ ((∀x) (¬ (Qyx))))
and finishing by dealing with the outermost ‘⇒’ symbols:
· ⇒ (⇒ (¬ (Px)) (¬ (Py)))Qxy
· ⇒ (¬ (⇒ ((∀y) (Py)) (¬ (¬ (Px))))) (¬ ((∀x) (¬ (Qyx))))
We have essentially completed the transformation of our formulae in Lmath into formulae in L.
It now remains to remove all brackets present, since, in the setting of L, these are unnecessary (and,
in any case, not allowed).
So, finally, as formulae in L, the given elements of Lmath become:
· ⇒⇒ ¬Px¬PyQxy
· ⇒ ¬ ⇒ ∀yPy¬¬Px¬∀x¬Qyx
Overall:
· ((Px) ∨ (¬ (Py)))⇒ Qxy corresponds to ⇒⇒ ¬Px¬PyQxy
· (((∀y) (Py)) ∧ (¬ (Px)))⇒ ((∃x) (Qyx)) corresponds to ⇒ ¬ ⇒ ∀yPy¬¬Px¬∀x¬Qyx
2
2. (a) (i) We form a semantic tableau starting from ¬ (((α ∨ β)⇔ (¬γ))⇒ ((¬γ) ∧ β)).
¬(((α ∨ β)⇔ (¬γ))⇒ ((¬γ) ∧ β))
(α ∨ β)⇔ (¬γ) , ¬((¬γ) ∧ β)
ggggg
ggggg
ggggg
ggggg
ggg
¬¬γ ¬β
SSSS
SSSS
SSSS
SSSS
SSSS
SSSS
SSSS
SSSS
SSSS
SSSS
γ
lll
lll
lll
lll
lll
α ∨ β , ¬γ ¬(α ∨ β) , ¬¬γ α ∨ β , ¬γ
UUUU
UUUU
UUUU
UUUU
UUUU
UU
¬(α ∨ β) , ¬¬γ
closed ¬α , ¬β α β ¬α , ¬β
γ open closed γ
open open
There exist open branches.
Since there exist open branches, we deduce that ((α ∨ β)⇔ (¬γ)) ⇒ ((¬γ) ∧ β) is not a
tautology.
By considering the open branches, we may describe every (type of) valuation, v, for which the
original proposition fails to be true:
· If v(α) = 0 and v(β) = 0 and v(γ) = 1, then v (((α ∨ β)⇔ (¬γ))⇒ ((¬γ) ∧ β)) = 0
· If v(α) = 1 and v(β) = 0 and v(γ) = 0, then v (((α ∨ β)⇔ (¬γ))⇒ ((¬γ) ∧ β)) = 0
3
(ii) We form a semantic tableau starting from ¬ (((¬γ) ∧ β)⇒ ((α ∨ β)⇔ (¬γ))).
¬(((¬γ) ∧ β)⇒ ((α ∨ β)⇔ (¬γ)))
(¬γ) ∧ β , ¬ ((α ∨ β)⇔ (¬γ))
¬γ , β
ggggg
ggggg
ggggg
ggggg
gg
VVVV
VVVV
VVVV
VVVV
VVVV
V
¬ (α ∨ β) , ¬γ α ∨ β , ¬¬γ
¬α , ¬β γ
NNN
NNN
NNN
NNN
NNN
closed α β
closed closed
All branches are closed.
Hence, the proposition ((¬γ) ∧ β)⇒ ((α ∨ β)⇔ (¬γ)) is a tautology.
4
(b) The given proposition is not a theorem.
We may start by noting the Completeness Theorem for propositional logic, which states that, for a
set of propositions S and a proposition δ:
S |= δ if and only if S ⊢ δ
As a result, we may conclude that it holds that |= ((α ∨ β)⇔ (¬γ))⇔ ((¬γ) ∧ β) if and only if it
holds that ⊢ ((α ∨ β)⇔ (¬γ)) ⇔ ((¬γ) ∧ β) holds, i.e. that ((α ∨ β)⇔ (¬γ)) ⇔ ((¬γ) ∧ β) is
a theorem if and only if it is a tautology.
Let us now try to show that the given proposition is not a tautology.
By examining the structure of the propositions present in this proposition and/or our solution to part
(a), we may describe a (type of) valuation for which the proposition is false.
For example, the given proposition is false for any valuation v for which
v (α) = 0 and v (β) = 0 and v (γ) = 1
For any such valuation v, v (α ∨ β) = 0 and v (¬γ) = 0. Therefore, v ((α ∨ β)⇒ (¬γ)) = 1 and
v ((¬γ)⇒ (α ∨ β)) = 1, so that v ((α ∨ β)⇔ (¬γ)) = 1.
Similarly, for any such valuation v, v (¬γ) = 0 and v (β) = 0, so that v ((¬γ) ∧ β) = 0.
Then, since v ((α ∨ β)⇔ (¬γ)) = 1 and v ((¬γ) ∧ β) = 0, we may deduce that
v (((α ∨ β)⇔ (¬γ))⇒ ((¬γ) ∧ β)) = 0 and v (((¬γ) ∧ β)⇒ ((α ∨ β)⇔ (¬γ))) = 1
so that
v (((α ∨ β)⇔ (¬γ))⇔ ((¬γ) ∧ β)) = 0
i.e. so that ((α ∨ β)⇔ (¬γ))⇔ ((¬γ) ∧ β) is not a tautology.
Hence, overall, we may conclude that the proposition ((α ∨ β)⇔ (¬γ))⇔ ((¬γ) ∧ β) is (also) not
a theorem, as indicated above.
5
3. (a) Consider {α⇒ ((¬β)⇒ γ) , α⇒ ((¬β)⇒ (¬γ))} ⊢ α⇒ β.
By the Deduction Theorem, to show {α ⇒ ((¬β)⇒ γ) , α ⇒ ((¬β)⇒ (¬γ))} ⊢ α ⇒ β, it is
due to suffice to give a proof of {α ⇒ ((¬β)⇒ γ) , α ⇒ ((¬β)⇒ (¬γ)) , α} ⊢ β; we do so
below:
1. α⇒ ((¬β)⇒ γ) Hypothesis
2. α⇒ ((¬β)⇒ (¬γ)) Hypothesis
3. α Hypothesis
4. ((¬β)⇒ (¬γ))⇒ (((¬β)⇒ γ)⇒ β) Axiom 3
5. (¬β)⇒ γ Modus ponens on lines 1, 3
6. (¬β)⇒ (¬γ) Modus ponens on lines 2, 3
7. ((¬β)⇒ γ)⇒ β Modus ponens on lines 4, 6
8. β Modus ponens on lines 5, 7
Hence, the original syntactic implication, {α ⇒ ((¬β)⇒ γ) , α ⇒ ((¬β)⇒ (¬γ))} ⊢ α ⇒ β,
holds.
(b) The following is a direct proof of the given syntactic implication:
1. (α⇒ (¬β))⇒ (γ ⇒ α) Hypothesis
2. ¬β Hypothesis
3. γ ⇒ (α⇒ δ) Hypothesis
4. (γ ⇒ (α⇒ δ))⇒ ((γ ⇒ α)⇒ (γ ⇒ δ)) Axiom 2
5. (γ ⇒ α)⇒ (γ ⇒ δ) Modus ponens on lines 3, 4
6. (¬β)⇒ (α⇒ (¬β)) Axiom 1
7. α⇒ (¬β) Modus ponens on lines 2, 6
8. γ ⇒ α Modus ponens on lines 1, 7
9. γ ⇒ δ Modus ponens on lines 5, 8
6
(c) A set S of propositions is consistent if there does not exist a proposition α (α ∈ L0) such that S ⊢ α
and S ⊢ ¬α; if there exists a proposition α such that S ⊢ α and S ⊢ ¬α, we may say that the set
S is inconsistent.
Let us consider the given set of propositions, {¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))}; we try to show
that we may prove each of the propositions β ⇒ ((γ ⇒ δ)⇒ (¬α)) and¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))
from the relevant set, i.e. that {¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))} ⊢ β ⇒ ((γ ⇒ δ)⇒ (¬α)) and
{¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))} ⊢ ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α))) (both) hold.
The following is a (direct) proof of β ⇒ ((γ ⇒ δ)⇒ (¬α)) from {¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))}:
1. ¬α Hypothesis
2. (¬α)⇒ ((γ ⇒ δ)⇒ (¬α)) Axiom 1
3. (γ ⇒ δ)⇒ (¬α) Modus ponens on lines 1, 2
4. ((γ ⇒ δ)⇒ (¬α))⇒ (β ⇒ ((γ ⇒ δ)⇒ (¬α))) Axiom 1
5. β ⇒ ((γ ⇒ δ)⇒ (¬α)) Modus ponens on lines 3, 4
Similarly, the following is a (direct) proof of the proposition ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α))) from the
given set propositions, i.e. from {¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))}:
1. ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α))) Hypothesis
Hence, it holds that {¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))} ⊢ β ⇒ ((γ ⇒ δ)⇒ (¬α)) and it also
holds that {¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))} ⊢ ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α))).
So, we may conclude that the given set of propositions, {¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))}, is not
consistent, i.e. that {¬α , ¬ (β ⇒ ((γ ⇒ δ)⇒ (¬α)))} is inconsistent.
7
4. (a) Let us first define an appropriate L(Π,Ω), i.e. sets of predicates and functionals we will use:
Π = {=,≤} , Ω = {A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11}
such that each of the predicate symbols ‘=’ and ‘≤’ is a binary predicate symbol, and such that each
of the functional symbols ‘A1’, ‘A2’, ‘A3’, ‘A4’, ‘A5’, ‘A6’, ‘A7’, ‘A8’, ‘A9’, ‘A10’, ‘A11’ is a 0-ary
functional symbol.
Then, we may use the theory consisting of the following sentences:
· (∀x)(x ≤ x)
· (∀x)(∀y)(((x ≤ y) ∧ (y ≤ x))⇒ (x = y))
· (∀x)(∀y)(∀z)(((x ≤ y) ∧ (y ≤ z))⇒ (x ≤ z))
· (∀x)(x = x)
· (∀x)(∀y) ((x = y)⇒ (y = x))
· (∀x)(∀y)(∀z) (((x = y) ∧ (y = z))⇒ (x = z))
· (∀x)(∀y)(∀z) ((x = y)⇒ ((z ≤ x)⇒ (z ≤ y)))
· (∀x)(∀y)(∀z) ((x = y)⇒ ((x ≤ z)⇒ (y ≤ z)))
· (∀x)((x = A1) ∨ (x = A2) ∨ (x = A3) ∨ (x = A4) ∨ (x = A5) ∨ (x = A6)
∨ (x = A7) ∨ (x = A8) ∨ (x = A9) ∨ (x = A10) ∨ (x = A11))
· ¬((∃x)(∀y)(x ≤ y))
8
(b) Let us first define an appropriate L(Π,Ω), i.e. sets of predicates and functionals we will use:
Π = {∼,=} , Ω = ∅
such that ‘∼’ and ‘=’ are binary predicate symbols.
Then, we may use the theory consisting of the following sentences:
· (∀x)(¬(x ∼ x))
· (∀x)(∀y)((x ∼ y)⇒ (y ∼ x))
· (∀x) (x = x)
· (∀x)(∀y) ((x = y)⇒ (y = x))
· (∀x)(∀y)(∀z) (((x = y) ∧ (y = z))⇒ (x = z))
· (∀x)(∀y)(∀z) ((x = y)⇒ ((z ∼ x)⇒ (z ∼ y)))
· (∀x)(∀y)(∀z) ((x = y)⇒ ((x ∼ z)⇒ (y ∼ z)))
· (∀x)(¬((∃y)(∃z)(∃w)((x ∼ y)∧(x ∼ z)∧(x ∼ w)∧(¬(y = z))∧(¬(y = w))∧(¬(z = w)))))
9
5. (a) (i) Below is a diagrammatic description of a register machine that computes
f1 : N0 → N0 , f1(m) = 8
S1R1−1 55

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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 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