I converting conditional statements into logical notation

[dabige1010]

i need to covert the following conditional statements into logical notation using propositional connectives and quantifiers:

a) A has at most one element


b)A is a singleton


c)ø ∈ A

you don't have to give me the answers, just help me get started or give me some hints

 

[cristo]

i need to covert the following conditional statements into logical notation using propositional connectives and quantifiers:

a) A has at most one element

Think of the cardinality of A.

b)A is a singleton

What is a singleton? Suppose A has two elements; what can you say about these elements?

c)ø ∈ A

This says "the empty set is a member of A." This doesn't make sense, to me; don't you mean "the empty set is a subset of A?"

you don't have to give me the answers, just help me get started or give me some hints[/QUOTE]

 

[dabige1010]

this is what I've come up with:

a) ∀x(x ∈ A → (x⇔ø v x ⇔ n))

b) ∀z(z ∈ A ⇔ z = x)

C) i didnt mistype, "ø ∈ A" is what the question said. i guess it's just a typo by the prof.

let me know what you think of the two answers i do have though.

thanks a lot!

 

[CRGreathouse]

ø ∈ A is quite sensible; it's used in the canonical set-theoretic construction of Peano arithmetic, for example. But I'm not sure what you'd need to do to rewrite it.

a) ∀x(x ∈ A → (x⇔ø v x ⇔ n))

b) ∀z(z ∈ A ⇔ z = x)

These have free variables, which I don't think you want. For the first one, I'd expect something like ∃n∀x (x ∈ A → x=n). Also, I'm not at all sure what you intend by "x⇔ø", which is surely not the same as your use of the double arrow in the second formula.

 

[cristo]

ø ∈ A is quite sensible; it's used in the canonical set-theoretic construction of Peano arithmetic, for example.

Fair enough. So; what does it mean?

 

[CRGreathouse]

Fair enough. So; what does it mean?

"The empty set is a member of A", what else? You might use the following definitions for numbers, for example:

0 = ø
S(n) = n U {n}

So that
1 = {ø} U ø = {0}
2 = {0} U {{0}} = {0, {0}} = {0, 1}
3 = {0, 1} U {{0, 1}} = {0, 1, 2}
. . .

"ø is a subset of A" is true for all sets A, but "ø is a member of A" is true for only some A. "ø ∈ ø" is false, for example; nothing is in the empty set, not even the empty set.