Originally Posted by
Jhevon lets work through these. remember the notations used, what they mean and how sets are defined
ok, so $\displaystyle A = \{ x \in \mathbb{Z} \mid -1 \le x \le 2 \} = \{ \text{The set of all integers between -1 and 2 inclusive} \}$
once you are able to interpret what the symbols mean, it is not so hard, right? now, lets write out the set
$\displaystyle A = \{ -1, 0, 1, 2 \}$
and $\displaystyle B = \{ 2x - 3 \mid x \in A \}$
that means we evaluate the expression 2x - 3 for all the elements of A
so, not to confuse you, but say $\displaystyle f(x) = 2x - 3$
then, $\displaystyle B = \{ f(-1), f(0), f(1), f(2) \}$
so, $\displaystyle B = \{ -5, -3, -1, 1 \}$
now, finally, $\displaystyle C = \left \{ x \in \mathbb{R} ~ \bigg| ~x = \frac ab, ~a \in A, ~b \in B \right \}$
so, you take each element of A and divide it by all the elements of B, one by one until you exhaust the elements of A. thus,
$\displaystyle C = \left \{ \frac {-1}{-5}, \frac {-1}{-3}, \frac {-1}{-1}, \frac {-1}{1}, 0, \frac {1}{-5}, \frac {1}{-3}, \frac {1}{-1}, \frac {1}{1}, \frac {2}{-5}, \frac {2}{-3}, \frac {2}{-1}, \frac {2}{1} \right \}$
$\displaystyle \Rightarrow C = \left \{ \frac 15, \frac 13, -1, 0, 1, - \frac 15, - \frac 13 , - \frac 25, - \frac 23, -2, 2 \right \}$
and you can write those in order if you like
Now, can you do the questions? i will remind you of the definitions
Let $\displaystyle X$ and $\displaystyle Y$ be sets. Then,
$\displaystyle X \cup Y = \{ x \mid x \in X \mbox{ or } x \in Y \}$
$\displaystyle X \times Y = \{ (x,y) \mid x \in X,~ y \in Y \}$
$\displaystyle X \backslash Y = \{x \mid x \in X \mbox{ and } x \notin Y \}$
$\displaystyle X \Delta Y = (X \backslash Y) \cup (Y \backslash X)$ (symmetric difference)