Chapter 6.4, Problem 1TY

In the comparison between the structure of the set of statement forms and the set of subsets of a universal set, the or operatioln V corresponds to --------, the and operation ? corresponds to -----, a tautology t corresponds to -------- a contradiction c corresponds to ----------, and the negation operation, denoted ?, corresponds to -------

To determine

To fill in the blanks of the given statement.

Answer to Problem 1TY

In the comparison between the structure of the set of statement forms and the set of subsets of a universal set, the $or$ operation $\vee$ corresponds to $\text{the}\text{operation}\text{of}\text{union}\cup$, the $and$ operation $\wedge$ corresponds to $\text{the}\text{operation}\text{of}\text{intersection}\cap$, a tautology $\text{t}$ corresponds to $\text{a}\text{universal}\text{set}U$, a contradiction $c$ corresponds to $\text{the}\text{empty}\text{set}\varnothing$ and the negation operation, denoted $\sim$, corresponds to $\text{the}\text{operation}\text{of}\text{complementation,}\text{denoted}\text{by}\text{using}\text{the}\text{superscript}c$.

Explanation of Solution

Given:

The given statement is “In the comparison between the structure of the set of statement forms and the set of subsets of a universal set, the $or$ operation $\vee$ corresponds to ______, the $and$ operation $\wedge$ corresponds to a tautology $\text{t}$ corresponds to, a contradiction $c$ corresponds toand the negation operation, denoted $\sim$, corresponds to.”

The logical equivalences of set operations are as shown below.

For $\text{the}\text{operation}\text{of}\text{union}\cup$, the logical equivalence is the $or$ operation $\vee$.

For $\text{the}\text{operation}\text{of}\text{intersection}\cap$, the logical equivalence isthe $and$ operation $\wedge$.

For $\text{a}\text{universal}\text{set}U$, the logical equivalence isa tautology $\text{t}$.

For $\text{the}\text{empty}\text{set}\varnothing$, the logical equivalence isa contradiction $c$.

And for the operation of complementation, the logical equivalence is the negation operation.

Hence, In the comparison between the structure of the set of statement forms and the set of subsets of a universal set, the $or$ operation $\vee$ corresponds to $\text{the}\text{operation}\text{of}\text{union}\cup$, the $and$ operation $\wedge$ corresponds to $\text{the}\text{operation}\text{of}\text{intersection}\cap$, a tautology $\text{t}$ corresponds to $\text{a}\text{universal}\text{set}U$, a contradiction $c$ corresponds to $\text{the}\text{empty}\text{set}\varnothing$ and the negation operation, denoted $\sim$, corresponds to $\text{the}\text{operation}\text{of}\text{complementation,}\text{denoted}\text{by}\text{using}\text{the}\text{superscript}c$.