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soopo
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Homework Statement
Let G be a group, and let H and K be two subgroups of G.
Then the union of H and K is a subgroup of G such that the intersection of H and K is an empty set.
Can you visualize the product hk in a venn diagram for all h
which belongs to H and for all k which belongs to K?
The Attempt at a Solution
You cannot because:
Take a point h at the top of H group where derivate of the "circle" is zero.
Take similarly a point k at the top of K.
Let k=(-1, 10) and h=(1, 10).
The product kh is
10 * 10 = 100
in y-direction, while
-1 * 1 = -1
in x-direction.
Thus, we get a point kh = (-1,100)
which does NOT belong to H nor K.
This suggests me that the set of the union H and K is greater than than
the high-school presentation of the union.
It seems that we cannot use venn diagrams in such problems.