Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C.
Show that B = C.
you can easily prove oit with a Venn diagram.
If not: A�¾B=A+B-A�¿B=A+B-A�¿C
AUC=A+C-A�¿C
As AUB=AUC then A+B-A�¿C=A+C-A�¿C hence; B=C
To show that B = C, we need to demonstrate that every element in B is also in C and vice versa. Here's how you can prove it using set theory:
1. Start with the given information: A ∪ B = A ∪ C and A ∩ B = A ∩ C.
2. Recall that the union of two sets A and B, denoted by A ∪ B, represents the set of elements that are in A, B, or both.
3. Likewise, the intersection of two sets A and B, denoted by A ∩ B, represents the set of elements that are common to both A and B.
4. Using the given information, A ∪ B = A ∪ C implies that any element that is in A is also in either B or C, or both.
5. Similarly, A ∩ B = A ∩ C implies that any element that is in both A and B is also in both A and C.
6. Consider an arbitrary element x that belongs to B. This means that x is either in A or B, or both. Since A ∪ B = A ∪ C, it follows that x must also be in C.
7. Next, consider an arbitrary element y that belongs to C. This means that y is either in A or C, or both. Again, since A ∪ B = A ∪ C, y must also be in B.
8. Since every element in B is also in C and every element in C is also in B, we conclude that B = C.
By showing that every element in B is also in C and vice versa, we have proven that B = C.