Im taking a mathematical proofs class and im thrown off on what the X's mean in this problem ! Any Advice?
Let U={1,2,3,4} be the universal set, and let A={1,2,3}. Determine the set S={X e P(A) : lXl =the complement of lXl} by listing its elements
Im taking a mathematical proofs class and im thrown off on what the X's mean in this problem ! Any Advice?
Let U={1,2,3,4} be the universal set, and let A={1,2,3}. Determine the set S={X e P(A) : lXl =the complement of lXl} by listing its elements
Sets contain elements. There is no rule that says these elements can't be sets themselves.
The power set P(A), is defined as a set whose elements are all the possible subsets of set A. Hence, each and every element of P(A) is a set itself.
When elements are not sets, we use lower case letters to denote them. When they are sets, then we use upper case letters to describe them.
So X is just a set, which is a member of the set P(A). And hence the elements of X are elements of A. X is a subset of A.
So you are being asked to find the set $\displaystyle S = \{X \in P(A) : |X| = |X^c|\} $
This translates, in english as:
Find S, which is the set of all sets X, which are members of the power set of A, such that the cardinality of X is equal to the cardinality of the complement of X.