F = { equilateral triangle}
and
G = { isosceles triangle }
a) is F ⊂ G ???
b) what is F ∩ G
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Let's look at the definitions: isosceles triangle: at least two sides are the same length equilateral triangle: all three sides are the same length Now, if we have an equilateral triangle, say with sides of length 5, 5, and 5, is there any way for that to NOT be an isosceles triangle? No, because if all three sides are the same, then at least two of them will be the same. So every equilateral triangle will also be an isosceles triangle. Is the reverse true? No, because we can easily make an isosceles triangle with sides of length 5, 5, and 6; or 5, 5, and 9; and so on. So if we imagine all the triangles that there could ever be, some of them will be isosceles, +---------------------------------+ | triangles | | | | +----------------------+ | | | isosceles triangles | | | | | | | +----------------------+ | | | +---------------------------------+ and some will be equilateral; but every equilateral triangle will also be isosceles, just as every isosceles triangle is also a triangle: +---------------------------------+ | triangles | | | | +----------------------+ | | | isosceles triangles | | | | | | | | +---------------+ | | | | | equilateral | | | | | | triangles | | | | | +---------------+ | | | | | | | +----------------------+ | | | +---------------------------------+
Let's look at the definitions: isosceles triangle: at least two sides are the same length equilateral triangle: all three sides are the same length Now, if we have an equilateral triangle, say with sides of length 5, 5, and 5, is there any way for that to NOT be an isosceles triangle? No, because if all three sides are the same, then at least two of them will be the same. So every equilateral triangle will also be an isosceles triangle. Is the reverse true? No, because we can easily make an isosceles triangle with sides of length 5, 5, and 6; or 5, 5, and 9; and so on. So if we imagine all the triangles that there could ever be, some of them will be isosceles, +---------------------------------+ | triangles | | | | +----------------------+ | | | isosceles triangles | | | | | | | +----------------------+ | | | +---------------------------------+ and some will be equilateral; but every equilateral triangle will also be isosceles, just as every isosceles triangle is also a triangle: +---------------------------------+ | triangles | | | | +----------------------+ | | | isosceles triangles | | | | | | | | +---------------+ | | | | | equilateral | | | | | | triangles | | | | | +---------------+ | | | | | | | +----------------------+ | | | +---------------------------------+
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