Project: More on \(\oplus\) and \(\times\)

David M. Clark and W. Ted Mahavier

( Chapter 3: W. Ted Mahavier < E.L. May < K.M. Shannon ¹ )

If each of \(X\) and \(Y\) is a person, then the relation \(X < Y\) indicates that the notes originated with \(Y\) and were subsequently modified by \(X\text{,}\) who takes full responsibility for the current version. While \(Y\) is credited with the genesis of the notes, s/he makes no claim to the accuracy of the current version which may or may not reflect her/his original version.

Section Project: More on \(\oplus\) and \(\times\)

For the following statements, either prove that it is true using the set equality axiom, or prove that it is false by exhibiting a counterexample.

Problem 1.31.

\(\A \oplus (\B \cap \C)=(\A \oplus \B) \cap (\A \oplus \C)\)

Problem 1.32.

\(\A \cap (\B \oplus \C) = (\A \cap \B) \oplus (\A \cap \C)\)

Problem 1.33.

\(\sim (\A \oplus \B) = \sim \A \oplus \sim \B\)

Problem 1.34.

\(\A \oplus \B = \sim \A \oplus \sim \B\)

Problem 1.35.

\(\A \cup (\B \oplus \C) = (\A \cup \B) \oplus (\A \cup \C)\)

Problem 1.36.

\(\A\times(\B\cup \C) = (\A\times \B) \cup (\A\times \C)\)

Problem 1.37.

\(\A\cup(\B\times \C) = (\A\cup \B) \times (\A\cup \C)\)