Find true or false:
if `A sube B ` then `A cup B = A `.

Answer

To determine whether the statement "If A is a subset of B, then A ∪ B = A" is true or false, we will analyze the definitions of subset and union step by step.

### Step 1: Understand the Definitions
- A **subset** (denoted as A ⊆ B) means that every element of set A is also an element of set B.
- A **proper subset** (denoted as A ⊂ B) means that A is a subset of B, and there exists at least one element in B that is not in A.
- The **union** of two sets A and B (denoted as A ∪ B) is the set that contains all elements that are in A, in B, or in both.

**Hint:** Remember the definitions of subset and union as they are crucial for understanding the relationship between the sets.

### Step 2: Analyze the Given Statement
The statement claims that if A is a subset of B, then A ∪ B = A.

1. If A is a subset of B, then all elements of A are included in B.
2. When we take the union of A and B (A ∪ B), we are combining all elements from both sets.

**Hint:** Think about what happens when you combine the elements of both sets.

### Step 3: Consider an Example
Let’s take an example to clarify:
- Let A = {1, 2} and B = {1, 2, 3}.
- Here, A is a proper subset of B (A ⊂ B).

Now, calculate A ∪ B:
- A ∪ B = {1, 2} ∪ {1, 2, 3} = {1, 2, 3}.

Now, compare A ∪ B with A:
- A = {1, 2} and A ∪ B = {1, 2, 3}.

**Hint:** Use specific sets to visualize the union and see if it equals A.

### Step 4: Conclusion
From our example, we see that A ∪ B = {1, 2, 3} which is not equal to A = {1, 2}. Therefore, the statement "If A is a subset of B, then A ∪ B = A" is **false**.

**Final Answer:** False.