Given suitable examples of sets A,B,C,show that:
A∩B=A∩C, B≠C
A∪B=A∪C, B≠C
To show that A∩B=A∩C, B≠C, we need to find suitable examples of sets A, B, and C.
Example:
Let's consider the following sets:
A = {1, 2, 3}
B = {1, 2}
C = {1, 3}
1. A∩B:
The intersection of sets A and B can be found by identifying the elements that are common to both sets. In this case, the elements 1 and 2 are present in both A and B, so A∩B = {1, 2}.
2. A∩C:
Similarly, the intersection of sets A and C can be found by identifying the elements that are common to both sets. In this case, the element 1 is present in both A and C, so A∩C = {1}.
From the above, we can see that A∩B = A∩C. Both intersections contain the element 1.
3. B≠C:
To prove that B is not equal to C, we need to demonstrate that they have at least one element that is different. In this case, B = {1, 2} and C = {1, 3}. The elements 2 and 3 are present in either B or C but not in both, so B is not equal to C.
Therefore, the suitable example of sets A, B, and C demonstrates that A∩B=A∩C, B≠C.
Now let's show that A∪B=A∪C, B≠C.
1. A∪B:
The union of sets A and B can be found by combining all the unique elements from both sets. In this case, the elements 1, 2, and 3 are present in either A or B or both, so A∪B = {1, 2, 3}.
2. A∪C:
Similarly, the union of sets A and C can be found by combining all the unique elements from both sets. In this case, the elements 1 and 3 are present in either A or C or both, so A∪C = {1, 2, 3}.
From the above, we can see that A∪B = A∪C. Both unions contain the elements 1, 2, and 3.
3. B≠C:
To prove that B is not equal to C, we need to demonstrate that they have at least one element that is different. In this case, B = {1, 2} and C = {1, 3}. The element 2 is present only in B, while the element 3 is present only in C, so B is not equal to C.
Therefore, the suitable example of sets A, B, and C demonstrates that A∪B = A∪C, B≠C.