Conncections Between Set Theory and Logic
S. F. Ellermeyer February 28, 2014
Suppose that is a non—empty set and suppose that { ()}∈ is a family of statements with domain . Then for each ∈ , the statement
() is either true or false (and not both). We will define to be the set of all values of ∈ for which () is true. Thus
={ ∈ | () is true} .
This means that the complement of (with regarded as the universal set) is
={ ∈ | () is false} .
For example, suppose we let = {1 2 3 4 5} and for each ∈ let
() : 2− 2 10.
Then (1), (2) and (3) are true and (4) and (5) are false. Thus
= {1 2 3} and = {4 5}. This is illustrated in the Venn diagram below.
1 Disjunctions and Conjunctions
If { ()}∈ and { ()}∈ are two families of statements with domain , then for each ∈ we can form the compound statements () ∨ (),
()∧ (), () =⇒ () and many other more complicated compound statements. Let us see how we can view such statements in terms of sets.
First consider the disjunction
()∨ () : () is true or () is true.
With our convention that
={ ∈ | () is true}
and
={ ∈ | () is true} , we see that
∪ = { ∈ | () is true or () is true} = { ∈ | () ∨ () is true}
Likewise we see that for the conjunction
()∧ () : () is true and () is true, we have
∩ = { ∈ | () ∧ () is true} .
As an example, suppose that = {1 2 3 4 5} and suppose that
() : 2− 2 10
() : (− 3) ( − 5) = 0.
In this case we have
= { ∈ | () is true} = {1 2 3}
={ ∈ | () is true} = {3 5}
= { ∈ | ˜ () is true} = {4 5}
={ ∈ | ˜ () is true} = {1 2 4}
∪ = { ∈ | () ∨ () is true} = {1 2 3 5}
∩ = { ∈ | () ∧ () is true} = {3} . This example is illustrated in the Venn diagram below.
2 Implications
If { ()}∈ and { ()}∈ are two families of statements with domain
, then for each ∈ , the implication () =⇒ () is false only for those values of for which () is true and () is false. Otherwise
() =⇒ () is true. We thus see that
{ ∈ | () =⇒ () is false} = { ∈ | () is true and () is false}
={ ∈ | () is true and ˜ () is true}
= ∩ .
This means that
{ ∈ | () =⇒ () is true} = ∩ .
By one of the DeMorgan Laws,
∩ = ∪ = ∪ .
Therefore
∪ = { ∈ | () =⇒ () is true} .
As an example, suppose that = {1 2 3 4 5} and suppose that
() : 2− 2 10
() : (− 3) ( − 5) = 0.
In this case we have
= { ∈ | () is true} = {1 2 3}
={ ∈ | () is true} = {3 5}
= { ∈ | ˜ () is true} = {4 5}
∪ = { ∈ | () =⇒ () is true} = {3 4 5} .
In the Venn diagram below, the shaded set is { ∈ | () =⇒ () is true}.
3 Quantified Statements
Suppose now that { ()}∈ and { ()}∈ are two families of statements with domain and let us see how we can study quantified statements con- cerning { ()}∈ and { ()}∈ in terms of the associated sets as defined above.
First let us consider the quantified statement
∀ ∈ , () is true.
Since = { ∈ | () is true}, then we could simply write the above statement (in terms of sets) as
= .
Next consider the statement
∃ ∈ such that () is true.
This statement says that () must be true for at least one value of ∈ .
Therefore we could write this statement (in terms of sets) as
6= ∅.
As another example, consider the statement
@ ∈ such that both () and () are true.
This statement can be written as
∩ = ∅.
Next consider the statement
∀ ∈ , () ∧ () is false.
This is equivalent to the statement given in the previous example and thus can also be written as
∩ = ∅.
We conclude with a couple of interesting examples that show how to use this “set approach” to logic to answer questions of the type given in the homework problems in Section 2.10 of the textbook with which we struggled somewhat to understand.
Example 1 Does the fact that
()∧ () is false for all ∈ imply that
(˜ ()) =⇒ () is false for some ∈ ?
We will reformulate this question in terms of sets. The statement
()∧ () is false for all ∈ can be written as
∩ = ∅.
Now, recalling that
∪ = { ∈ | () =⇒ () is true} , we see that
∪ = { ∈ | (˜ ()) =⇒ () is true} .
Thus the statement
(˜ ()) =⇒ () is false for some ∈ can be written as
∪ 6= ∅.
Overall, we now see that the question that is being asked in this example is that if the fact that ∩ = ∅ implies that ∪ 6= ∅. We can see that the answer is no by drawing a simple Venn diagram in which ∩ = ∅ is true but ∪ 6= ∅ is false. Such a Venn diagram is shown below.
A specific example that goes along with the above diagram (in terms of logic) is: Let = {1 2 3} and let
() : = 1
() : (− 2) ( − 3) = 0.
Then the statement
()∧ () : = 1 and ( − 2) ( − 3) = 0
is indeed false for all ∈ but the statement
(˜ ()) =⇒ () : 6= 1 =⇒ ( − 2) ( − 3) = 0 is true for all ∈ .
Example 2 Does the fact that
()∨ () is false for some ∈ imply that
(˜ ()) =⇒ () is false for some ∈ ?
Reformulating the question in terms of sets, we see that the statement
()∨ () is false for some ∈ can be written as
∪ 6= ∅ and the statement
(˜ ()) =⇒ () is false for some ∈ can in fact also be written as
∪ 6= ∅.
Thus the fact that
()∨ () is false for some ∈ does imply that
(˜ ()) =⇒ () is false for some ∈ .
(In fact these two statements are logically equivalent: Each implies the other.)
