Introduction to Abstract Algebra/Problem set 1

Problem Set #1: Introduction to Abstract Algebra.

As you work through these problems, think about the logical steps you are using. You should know if your proof is correct or not if you have a reason for every step.

1: If$$B \subset A$$, prove $$A \cup B = A$$ and conversely

2: Prove $$( A \cap B ) \cap C = A \cap ( B \cap C )$$

3: Prove $$A \cup ( B \cap C ) = ( A \cup B ) \cap ( A \cup C )$$

4: Define the set C' to be the complement of the set C relative to a defined universal set U for which $$C \subset U$$ such that $$\forall c \in C, c \notin C'$$. Prove that $$( A \cap B )' = A' \cup B'$$ and $$( A \cup B )' = A' \cap B'$$.