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Problem 11.4. Let Þ(x, y) be a formula with at most two free vari- ables x and y. Suppose also that no occurrence of x or y in is bounded by a quantifier. Prove that the following sentences are true. (1) (Vx)(Vy)Þ(x, y) ⇒ (Vx)Þ(x,x) (2) (3x)Þ(x,x) ⇒ (3x) (‡y)Þ(x, y) (3) (3x)(Vy)Þ(x, y) ⇒ (Vy)(3x)Þ(x, y)

Problem 11.4. Let Þ(x, y) be a formula with at most two free vari- ables x and y. Suppose also that no occurrence of x or y in is bounded by a quantifier. Prove that the following sentences are true. (1) (Vx)(Vy)Þ(x, y) ⇒ (Vx)Þ(x,x) (2) (3x)Þ(x,x) ⇒ (3x) (‡y)Þ(x, y) (3) (3x)(Vy)Þ(x, y) ⇒ (Vy)(3x)Þ(x, y)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 32EQ
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Please provide a full proof for each sentence. 

Problem 11.4. Let Þ(x, y) be a formula with at most two free vari- ables x and y. Suppose also that no occurrence of x or y in D is bounded by a quantifier. Prove that the following sentences are true. (1) (Vx)(Vy)Þ(x, y) ⇒ (Vx)Þ(x,x) (2) (3x)Þ(x,x) ⇒ (3x)(y)Þ(x, y) (3) (3x)(Vy)Þ(x, y) ⇒ (Vy)(3x)Þ(x, y)
Transcribed Image Text:Problem 11.4. Let Þ(x, y) be a formula with at most two free vari- ables x and y. Suppose also that no occurrence of x or y in D is bounded by a quantifier. Prove that the following sentences are true. (1) (Vx)(Vy)Þ(x, y) ⇒ (Vx)Þ(x,x) (2) (3x)Þ(x,x) ⇒ (3x)(y)Þ(x, y) (3) (3x)(Vy)Þ(x, y) ⇒ (Vy)(3x)Þ(x, y)
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