Please answer any three quistions of these : (a) (c) (f) (g) (h) in clear hand w

Question

Please answer any three quistions of these : (a) (c) (f) (g) (h)

in clear hand writing or type it.

Thank you

3. This question involves formalizing some simple properties of sets in FOL. Consider the following three facts: No set is an element of itself. A set x is a subset of a set y iff every element of x is an element Something is an element of the union of two sets x and y iff it is an element of x or an element ofy (a) Represent the facts as sentences of FOL. As nonlogical symbols, use Sub(x,y to mean "x is a subset of y," Elt(e,x) to mean "e is an element of x," and u(x,y) to mean "the union of x and y." Instead of using a special predicate to assert that something is a set, you may simply assume that in the domain of discourse (assumed to be nonempty) everything is a set. Call the resulting set of sentences T. of the union of x and y union of x and y is equal to the union of y and x (b) Show using logical interpretations that T entails that x is a subset (c) Show using logical interpretations that T does not entail that the (d) Let A be any set. Show using logical interpretations that T entails that there is a set z such that the union of A and z is a subset of A (e) Does T entail that there is a set z such that for any set x the union of x and z is a subset ofx? Explain (f) Write a sentence that asserts the existence of singleton sets, that is, for any x, the set whose only element is x.is T with this sentence added. (g) Prove that Ti is not finitely satisfiable (again, assuming the domain is nonempty. Hint: In a finite domain, consider u, the object interpreted as the union of all the elements in the domain (h) Prove or disprove that T entails the existence of an empty set

Explanation / Answer

a). By the second definition sub(x,y) that means 'x' is a subset of 'y' iff every element of a set 'x' is a subset of set 'y'.

Elt(e,x) means 'e' is an element of 'x' therefore we can say that 'e' is an element of 'y'

u(x,y) i.e. union of x and y. Since 'x' is a subset of 'y'

u(x,y)= y

Hence T=y

b). As T = u(x,y) #which is equal to set 'y' itself

It is trivial that 'x' is a subset of 'y'

i.e sub(x,y)

This implies x is a subset of union of 'x' and 'y'

c). u(x,y) something is an element of union of x and y iff it is an element of x or y

Since 'e' is an element of 'x' hence it is in u(x,y)

Similarly 'e' is an element of u(y,x)

Hence T does not entail that union of x and y is equal to union of y and x.

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