snshine415
New member
- Joined
- Aug 31, 2010
- Messages
- 1
After creating the Venn diagrams for the symmetric difference of A and B and (A U B) - (B ?A) using the sets A = {1,2,3,4} and B = {3,4,5,6}, I came to the conclusion that the final diagrams were identical. Applying this to the example, the symmetric difference of A and B is (ex. {1,2,5,6}) which show as the shaded portion of the Venn diagram, leaving the intersecting (ex. {3,4}) un-shaded. as it is to include elements in both A and B.
Then as I drew the diagram for #2, The first portion (A U B) was the entirety of both sets, including elements of either A or B, so both circles- as well as the overlap were shaded (ex. {1,2,3,4,5,6}). As I took the next step to solve the second portion of the problem (B ?A), the shaded area was the intersection only (ex. {3,4}). Subtracting the two, my ending Venn diagram showed only the intersecting portion of (ex. {3,4}) again un-shaded, as the symmetric difference was again depicted with the shading of (ex. {1,2,5,6}).
It would so seem to me that the equation presented in #2 would be equivalent to finding the symmetric difference. As the symmetric difference = (A-B) U (B-A) which would include {1,2} for (A-B) and {5,6} for (B-A) as the union would include the elements of either but the difference would only include the elements that are in A but not B or B but not as diagramed the same as A. The second problem of (A U B) - (B ?A)where as {x?x ? A or x ? B} – {x?x ? B and x ? A} would again begin with the elements of both sets as implied by the union {1,2,3,4,5,6} and then go on to subtract the intersecting portion of the two sets {3,4} which left the remainder {1,2,5,6} equal to the symmetric difference.
What I am trying to ascertain is the reason as to why the two are identical and why the formula works with various elements of the set? Is there a simple explanation for this elementary problem?
Then as I drew the diagram for #2, The first portion (A U B) was the entirety of both sets, including elements of either A or B, so both circles- as well as the overlap were shaded (ex. {1,2,3,4,5,6}). As I took the next step to solve the second portion of the problem (B ?A), the shaded area was the intersection only (ex. {3,4}). Subtracting the two, my ending Venn diagram showed only the intersecting portion of (ex. {3,4}) again un-shaded, as the symmetric difference was again depicted with the shading of (ex. {1,2,5,6}).
It would so seem to me that the equation presented in #2 would be equivalent to finding the symmetric difference. As the symmetric difference = (A-B) U (B-A) which would include {1,2} for (A-B) and {5,6} for (B-A) as the union would include the elements of either but the difference would only include the elements that are in A but not B or B but not as diagramed the same as A. The second problem of (A U B) - (B ?A)where as {x?x ? A or x ? B} – {x?x ? B and x ? A} would again begin with the elements of both sets as implied by the union {1,2,3,4,5,6} and then go on to subtract the intersecting portion of the two sets {3,4} which left the remainder {1,2,5,6} equal to the symmetric difference.
What I am trying to ascertain is the reason as to why the two are identical and why the formula works with various elements of the set? Is there a simple explanation for this elementary problem?