Given below is an attempted proof of a result. Proof First,...

Given below is an attempted proof of a result. Proof First, we show that $A \subseteq(A \cup B)-B$. Let $x \in A$. Since $A \cap B=\emptyset$, it follows that $x \notin B$. Therefore, $x \in A \cup B$ and $x \notin B$; so $x \in(A \cup B)-B$. Thus $A \subseteq(A \cup B)-B$. Next, we show that $(A \cup B)-B \subseteq A$. Let $x \in(A \cup B)-B$. Then $x \in A \cup B$ and $x \notin B$. From this, it follows that $x \in A$. Hence $(A \cup B)-B \subseteq A$. (a) What result is being proved above? (b) What change (or changes) in this proof would make it better (from your point of view)?

Given below is an attempted proof of a result.
Proof First, we show that $A \subseteq(A \cup B)-B$. Let $x \in A$. Since $A \cap B=\emptyset$, it follows that $x \notin B$. Therefore, $x \in A \cup B$ and $x \notin B$; so $x \in(A \cup B)-B$. Thus $A \subseteq(A \cup B)-B$.
Next, we show that $(A \cup B)-B \subseteq A$. Let $x \in(A \cup B)-B$. Then $x \in A \cup B$ and $x \notin B$. From this, it follows that $x \in A$. Hence $(A \cup B)-B \subseteq A$.
(a) What result is being proved above?
(b) What change (or changes) in this proof would make it better (from your point of view)?

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Key Concepts

Set Operations

Understanding basic set operations such as union, difference, and intersection is crucial. The union of two sets combines all elements from both sets, while the set difference removes elements belonging to a specified set. These operations are the building blocks for many arguments and proofs in set theory.

Disjoint Sets

Disjoint sets have no elements in common; their intersection is empty. This property is essential in proofs where the absence of shared elements directly influences the set relationships, as it can simplify arguments by ruling out potential overlaps between sets.

Set Equality

Set equality asserts that two sets are the same if and only if every element of one set is an element of the other, and vice versa. This concept is fundamental in set theory and is typically demonstrated by proving mutual subset relations—showing that each set is contained within the other.

Double Inclusion Method

The double inclusion method is a proof technique where one establishes the equality of two sets by proving both that the first set is a subset of the second and that the second is a subset of the first. This method is widely used because it breaks the overall proof into two manageable parts, each addressing one direction of the containment.

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