Set Operations

Design algorithms for following set operations on sets A and $B$

$(1)$ Check whether A satisfies definition of set

$(2)$ Find cardinality of $A$

$(3)$ Element a belongs to $A$

Complement of A considering $A$ is a subset of universal set $U$

Union of A and $B$

Intersection of A and $B$

Set difference $A-B$

$(8)$ Check whether $A$ is a subset of $B$

Check whether $A$ is a superset of $B$

$(10)$ Cartesian Product $-AB$

Power set of $A-P(A)$

Provide time and space required by each of the algorithms.

Prove that the algorithms are correct.

Considering $U=\{a,$dots,$z\},A=\{a,f,l,q\}$ and $B=\{d,f,m,w\}$ give trace for each algorithms.