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Equality

Definition 3.2.6   Two sets $ A$ and $ B$ are equal if $ A \subset B$ and $ B \subset A$.

This is exactly the meaning of a sentence such that: ``the set of real solutions of the quadratic equation $ x^2-5x+6=0$ is $ \{2,3 \}$''. If the number $ a$ is a solution of the given equation, then $ a=2$ or $ a=3$; conversely, if $ a=2$ or $ a=3$, then $ a$ is a solution of the given quadratic equation. If two sets $ A$ and $ B$ are not equal, we denote $ A \neq B$. For example, $ \{ a,c,d,e \} \neq \{ a,b,c \}$. Note that it suffices to find one element who belongs to one set and not to the other to prove inequality of two sets.

root 2002-06-10