Hello Prof and Classmates!
An equivalent set is when two sets have the same AMOUNT of elements/member. An equal set is when set one and set two both contain the EXACT same elements/members. Therefore, equivalent means same amount of elements but not necessarily exactly the same, where equal means all elements are exactly the same, but necessarily in exact order.
Example: Equivalent set A={2,4,6} B={3,5,7} Equal set A={3,6,9} B={9,3,6}.
A subset is a set of elements that must have all of its elements belonging to an original set. A proper subset is a set of elements that must all of its elements belonging to the original set, but the original set must have an element that is not contained in the second set. Therefore, the difference between the a subset and proper subset is; in a subset the sets will be equal, but in a proper subset the set will not be equal.
Example: Subset A={6,7,8,9} B={6,7} Proper Subset A={h,o,m,e} B={h,o,m,e,s}
A complement of a set is when all of the elements in the union/universal set are not in the original set. The universal set is the set containing all elements of either both sets, or just of set one and set two.
Example: Complement of a set U={a,b,c,d,e,f,g,h} A={b,d,f} B={a,b,c,d,e} Union U={q,r,s,t,u,v,w,x,y,z} A={q,r,s,t} B={q,r,s,u,w} C={r,s,u,v}
Cardinality of a set is simply the amount of elements it contains. This function relates to the number of subsets of a set when you begin to use terms such as equal, equivalent, complement, union, and intersection.
Example: Set A = {1,3,5} Set B = {2,4,6}. Both sets have 3 elements, therefore, the cardinality is equal to 3. Also that makes the two sets equivalent since they both contain three elements. In order for the sets to be equal, they must contain the exact same elements.