For each of the following set expressions, say what set the expression denotes. In other words if the se....

Question

For each of the following set expressions, say what set the expression denotes. In other words if the set is finite, give an explicit listing of its elements.. Assume A = {2,3,4}

a) {x:x ∈ Z, -3.5 < x < 3.5}∩ {y:y ∈ Z, 2|y}

b) {x:x ∈ P(A), |x| = 2}

So for a) would I put something like... x such that x is a element of all integers between -3.5 and 3.5 AND y such that y is a element of all integers, but I don;t know how to interpret the | symbol. What does that mean? Am I going about this the correct way? And I have no idea how to do b)

Answer 1

b)

$|x|$ is the cardinality of $x$ i.e the number of elements.

as $\mathcal{P}(A) = \{\emptyset,\{2\},\{3\},\{4\},\{2,3\},\{2,4\},\{3,4\},\{2,3,4\}\}$

the set must contain of the three elements of $\mathcal{P}$ having cardinality $2$, i.e the entire set is:

$\{x : x \in \mathcal{P}(A), |x| = 2\} = \{\{2,3\},\{2,4\},\{3,4\}\}$