Understanding Set Difference (A − B)

The problem asks us to find the value of A − B, given two sets A and B. This operation is called the set difference. The set difference A − B (read as "A minus B") is the set containing all the elements that are in set A but are NOT in set B.

In simpler terms, you start with all the elements in set A and then remove any elements that are also present in set B. The elements remaining in set A after this removal form the set A − B.

We are given the following sets:

• Set A = $\{1, 3, 5, 7\}$
• Set B = $\{1, 4, 7\}$

Finding Elements in Set A but Not in Set B

To calculate A − B, we look at each element in set A and check if it is also in set B.

1. Consider the first element of set A, which is 1. Is 1 in set B? Yes, 1 is in set B. So, 1 is NOT in A − B.
2. Consider the second element of set A, which is 3. Is 3 in set B? No, 3 is NOT in set B. So, 3 IS in A − B.
3. Consider the third element of set A, which is 5. Is 5 in set B? No, 5 is NOT in set B. So, 5 IS in A − B.
4. Consider the fourth element of set A, which is 7. Is 7 in set B? Yes, 7 is in set B. So, 7 is NOT in A − B.

The elements from set A that are not present in set B are 3 and 5.

Therefore, the set A − B consists of these elements.

A − B = $\{3, 5\}$

Summary of the Set Difference Calculation

Let's summarize the steps to find the set difference A − B:

• List the elements of set A: $\{1, 3, 5, 7\}$
• List the elements of set B: $\{1, 4, 7\}$
• Identify elements that are in set A AND in set B (the intersection): $\{1, 7\}$
• Remove these common elements from set A.

Elements in A: 1, 3, 5, 7
Common elements to remove: 1, 7
Elements remaining in A after removal: 3, 5

So, A − B = $\{3, 5\}$. This result is a new set containing only the elements unique to set A when compared to set B.