{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math+150++F09-+Hwk+5+Solutions

# Math+150++F09-+Hwk+5+Solutions - SOLUTIONS TO HOMEWORK 5...

This preview shows pages 1–2. Sign up to view the full content.

SOLUTIONS TO HOMEWORK 5 MATH 150, FALL 09 Problem 1. Section 2.2/ Exercise 8. Suppose Σ is a set of formulas such that for all sentences τ , either Σ | = τ or Σ | = ¬ τ . Assume that A | = Σ . Show that for any τ , A | = τ iff Σ | = τ . For one of the directions, suppose that Σ | = τ . Then by definition of logical implication and since A | = Σ, it follows that A | = τ . For the other direction, suppose that A | = τ . Then A 6| = ¬ τ . By the definiton of logical implication and since A | = Σ, it follows that Σ 6| = ¬ τ . So, by the assumptions on Σ, we get that Σ | = τ . Problem 2. Section 2.2/ Exercise 9. Assume that the language has equality and a two-place predicate P . For each of the following find a sentence σ such that the model A satisfies σ iff the condition is met. (1) The universe of A has exactly two members. (2) P A is a function (from the universe of A to itself). (3) P A is a permutation of the universe of A (i.e. it is a one-to-one, onto function). (1) σ = “ x y ( ¬ ( x = y ) ∧ ∀ z ( z = x z = y ))” (2) σ = “ x y ( P ( x, y ) ∧ ∀ z ( P ( x, z ) y = z ))” (3) Let σ f = “ x y ( P ( x, y ) ∧∀ z ( P ( x, z ) y = z ))” and σ 11 = “ x y ( z ( P ( x, z ) P ( y, z )) x = y )” and σ o = “ y xP ( x, y )”. Then σ f says that P A is a function, σ 11

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Math+150++F09-+Hwk+5+Solutions - SOLUTIONS TO HOMEWORK 5...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online