Understanding Set Operations and Properties

This question asks us to evaluate the correctness of two statements regarding operations on two non-empty sets, A and B, specifically when the sets are equal (A = B).

Let's analyze each statement separately, keeping in mind that A and B are non-empty sets and the condition is A = B.

Analyzing Statement 1: A ∪ B = A ∩ B if A = B

Statement 1 claims that the union of sets A and B is equal to the intersection of sets A and B if A is equal to B.

• The union of two sets A and B, denoted by $A \cup B$, is the set of all elements that are in A, or in B, or in both.
• The intersection of two sets A and B, denoted by $A \cap B$, is the set of all elements that are common to both A and B.

Given the condition $A = B$, let's see what $A \cup B$ and $A \cap B$ become:

• If $A = B$, then $A \cup B$ is the union of A with itself, which is simply A. So, $A \cup B = A$.
• If $A = B$, then $A \cap B$ is the intersection of A with itself, which is also simply A. So, $A \cap B = A$.

Since both $A \cup B$ and $A \cap B$ are equal to A when $A = B$, it follows that $A \cup B = A \cap B$.

Therefore, Statement 1 is correct.

Analyzing Statement 2: A Δ B = ϕ if A = B

Statement 2 claims that the symmetric difference of sets A and B is equal to the empty set ($\varphi$) if A is equal to B.

• The symmetric difference of two sets A and B, denoted by $A \Delta B$, is the set of elements which are in either A or B, but not in their intersection. It can be defined as $(A \cup B) - (A \cap B)$ or $(A - B) \cup (B - A)$.
• The empty set, denoted by $\varphi$, is the set containing no elements.

Given the condition $A = B$, let's use the definition $A \Delta B = (A - B) \cup (B - A)$:

• If $A = B$, then $A - B$ is the set of elements in A but not in B. Since A and B have exactly the same elements, there are no elements in A that are not in B. So, $A - B = \varphi$.
• Similarly, if $A = B$, then $B - A$ is the set of elements in B but not in A. Since A and B have the same elements, there are no elements in B that are not in A. So, $B - A = \varphi$.

Now, let's find the union of these two results:

• $A \Delta B = (A - B) \cup (B - A) = \varphi \cup \varphi = \varphi$.

Alternatively, using the definition $A \Delta B = (A \cup B) - (A \cap B)$: We already found that if $A=B$, then $A \cup B = A$ and $A \cap B = A$. So, $A \Delta B = A - A = \varphi$.

In both cases, the symmetric difference $A \Delta B$ is the empty set when $A = B$.

Therefore, Statement 2 is correct.

Conclusion on Set Statements

Both Statement 1 ($A \cup B = A \cap B$ if $A = B$) and Statement 2 ($A \Delta B = \varphi$ if $A = B$) have been shown to be correct based on the definitions of set union, intersection, and symmetric difference when the sets A and B are equal.

Based on our analysis, both statements are correct.

Revision Table: Key Set Concepts

Additional Information: Properties of Sets

When working with sets, understanding basic properties and definitions is crucial. Here are a few more points:

• Set Equality: Two sets A and B are equal ($A = B$) if and only if they have exactly the same elements. This means every element in A is in B, and every element in B is in A.
• Non-empty Sets: The problem specifies that A and B are non-empty sets. This means they each contain at least one element. However, the results we derived ($A \cup B = A \cap B = A$ and $A \Delta B = \varphi$) when $A=B$ would also hold true even if A and B were the empty set ($\varphi$), as $\varphi = \varphi$. $\varphi \cup \varphi = \varphi$ and $\varphi \cap \varphi = \varphi$, so $\varphi \cup \varphi = \varphi \cap \varphi$. Also, $\varphi \Delta \varphi = (\varphi - \varphi) \cup (\varphi - \varphi) = \varphi \cup \varphi = \varphi$.
• Set Operations are Well-Defined: Union, intersection, and symmetric difference are fundamental operations in set theory and have clear definitions regardless of the contents of the sets.