4b. In the following exercises I give you the symbols that you will use for predicates or relations. Your task is to paraphrase the English sentences in quantified relational logic. For example, suppose you have the English sentence “Something is missing” (M, missing), then the paraphrase is ∃xMx.(i) All swans are white. (S, swan; W, white)(ii) Every swan is white. (S, swan; W, white)(iii) Each swan is white. (S, swan; W, white)(iv) Only white swans exist. (S, swan; W, white)(v) A swan is white. (S, swan; W, white) (The image is just formuals if your not familiar with this) We are now in a position do describe quantified relational logicfor one or two place relations only. Remember that to specify alogic I need to tell you (1) the formal symbols, (2) thetransformation rules, and (3) closure condition.The formal symbols are:· a, b, c, . . . , m, n, . . . as constant symbols for terms. A term isanything in a theory or language that can be given a propername 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, . . . ofalphabet as symbols for relations (on two placetheonly).- 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 thesymbols F, G and H.The transformation rules for quantified relational logic are:1. For any predicate symbol P and for any two place relationsymbol 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 andy are said to "free" variables because there are no quantifiers towhich they are bound.2. If F is a formula by Rule 1 and if x is a free variable in F, then3x F is a formula and Vx F is a formula. Rule 2 is known asbinding any free variable x in F by a quantifier.A formula F formed by either rule 1 and 2 is called an atomicformula.3. If F and G are atomic formulas, then -F, (F v G), (F ^ G),· G) and (F ↔ G) are complex formulas.(F4. If H is a complex formula, then the result of binding any freevariable in H is a formula.

Asper the guidelines I’m supposed to solve only three subparts of the given problem: Given that: (i)…

Answered: 4b. In the following exercises I give…

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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4b. In the following exercises I give you the symbols that you will use for predicates or relations. Your task is to paraphrase the English sentences in quantified relational logic. For example, suppose you have the English sentence “Something is missing” (M, missing), then the paraphrase is ∃xMx.

(i) All swans are white. (S, swan; W, white)

(ii) Every swan is white. (S, swan; W, white)

(iii) Each swan is white. (S, swan; W, white)

(iv) Only white swans exist. (S, swan; W, white)

(v) A swan is white. (S, swan; W, white)

(The image is just formuals if your not familiar with this)

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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4d. In the following exercises I give you the symbols that you will use for predicates or relations. Your task is to paraphrase the English sentences in quantified relational logic. For example, suppose you have the English sentence “Something is missing” (M, missing), then the paraphrase is ∃xMx.

(iv) Only white swans exist. (S, swan; W, white)

(v) A swan is white. (S, swan; W, white)

(The image is just formuals if your not familiar with this)