C is the circumcenter of isosceles triangle ABD with vertex angle ABD. Does the following proof correctly justify that triangles
ABE and DBE are congruent? 1. It is given that triangle ABD is an isosceles triangle, so segments AB and DB are congruent by the definition of isosceles triangle. 2. It is given that is the circumcenter of triangle ABD, making segment BE a median 3. By the definition of perpendicular, angles AEB and DEB are 90°, so triangles ABE and DEB are right triangles. 4. Triangles ABE and DEB share side BE making it congruent to itself by the reflexive property 5. Triangles ABE and DBE are congruent by HL. E There is an error in line 1, segments AB and BD are given to be congruent. There is an error in line 2, segment BE should be a perpendicular bisector. There is an error in line 4, segment BE is not a shared side. The proof is correct
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The inclusion/exclusion principle states that
That is, the union has as many members as the sum of the number of members of the individual sets, minus the number of elements contained in both sets (to avoid double-counting).
Therefore,
will have the most elements when the sets 
and 
are disjoint, i.e. 
, which would mean the most we can can in this case would be
(Note that
denotes the cardinality of the set .)
That is, the union has as many members as the sum of the number of members of the individual sets, minus the number of elements contained in both sets (to avoid double-counting).
Therefore,
(Note that