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Question 1  In this exercise use the transformation rules given in the picture uploaded, to verify that the following are formulas of quantified relational logic. (a) Ba  (b) ¬Bb  (c) ∃x¬Bx  (d) ∃x(Rx ∧ ¬Bx)  (e) ¬∀x(Rx → Bx)  (f) ((Rb ∧ ¬Bb) → ∃x(Rx ∧ ¬Bx))    (The image shows rules and formulas if your not familiar with this)

Question 1  In this exercise use the transformation rules given in the picture uploaded, to verify that the following are formulas of quantified relational logic. (a) Ba  (b) ¬Bb  (c) ∃x¬Bx  (d) ∃x(Rx ∧ ¬Bx)  (e) ¬∀x(Rx → Bx)  (f) ((Rb ∧ ¬Bb) → ∃x(Rx ∧ ¬Bx))    (The image shows rules and formulas if your not familiar with this)

Advanced Engineering Mathematics
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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Question 1 
In this exercise use the transformation rules given in the picture uploaded, to verify
that the following are formulas of quantified relational logic.
(a) Ba 
(b) ¬Bb 
(c) ∃x¬Bx 
(d) ∃x(Rx ∧ ¬Bx) 
(e) ¬∀x(Rx → Bx) 
(f) ((Rb ∧ ¬Bb) → ∃x(Rx ∧ ¬Bx)) 
 
(The image shows rules and formulas 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.
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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