If A ⊂ B, then A ∩ B = A ∪ B.sometimesalwaysnever

To determine whether the statement "If A ⊂ B, then A ∩ B = A ∪ B" is sometimes, always, or never true, we can break it down step by step.

Step 1: If A ⊂ B This means that A is a subset of B, which implies that every element in A is also in B.

Step 2: A ∩ B The intersection of A and B represents the set of elements that are common to both A and B.

Step 3: A ∪ B The union of A and B represents the set of all elements that are in either A or B, or both.

Now, let's analyze the statement:

If A ⊂ B, then A ∩ B = A ∪ B

If A is a subset of B, it means that every element in A is also in B. In this case, the intersection of A and B (A ∩ B) will be equal to A itself, because all the elements in A are already in B. Therefore, A ∩ B = A.

On the other hand, the union of A and B (A ∪ B) will include all the elements in A and all the elements in B, without any repetition. Since A is already a subset of B, adding B to A will not introduce any new elements. Therefore, A ∪ B = B.

So, in conclusion:

A ∩ B = A A ∪ B = B

Therefore, the statement "If A ⊂ B, then A ∩ B = A ∪ B" is always true.

To determine whether the statement "If A ⊂ B, then A ∩ B = A ∪ B" is sometimes, always, or never true, we can break it down step by step.

Step 1: If A ⊂ B This means that A is a subset of B, which implies that every element in A is also in B.

Step 2: A ∩ B The intersection of A and B represents the set of elements that are common to both A and B.

Step 3: A ∪ B The union of A and B represents the set of all elements that are in either A or B, or both.

Now, let's analyze the statement:

If A ⊂ B, then A ∩ B = A ∪ B

If A is a subset of B, it means that every element in A is also in B. In this case, the intersection of A and B (A ∩ B) will be equal to A itself, because all the elements in A are already in B. Therefore, A ∩ B = A.

On the other hand, the union of A and B (A ∪ B) will include all the elements in A and all the elements in B, without any repetition. Since A is already a subset of B, adding B to A will not introduce any new elements. Therefore, A ∪ B = B.

So, based on the analysis, we can conclude that if A ⊂ B, then A ∩ B = A and A ∪ B = B. Therefore, the statement "If A ⊂ B, then A ∩ B = A ∪ B" is always true.