Assignment:
Let A and B be sets. Show that A x B and B x A have equal cardinality by constructing an explicit bijection between the two sets. Then use the following proposition to prove that multiplication is commutative. (Let n, m be natural numbers. Then nxm=mxn)
Proposition: Cardinal arithmetic
a) Let X be a finite set, and let x be an object which is not an element of X. Then X U (union) {x} is finite and #(X U {x})= #(X)+1
b) Let X and Y be finite sets. Then X U Y is finite and # (X U Y) is less than or equal to #(X) + #(Y). If in addition X and Y are disjoint (i.e., X intersection Y = the empty set), then #(X U Y)= #(X) + #(Y)
c) Let X be a finite set, and let Y be a subset of X. Then Y is finite, and #(Y) is less than or equal to #(X), If in addition Y does not equal X (i.e. Y is a proper subset of X), then we have #(Y) is less than #(X)
d) If X is a finite set, and f:X-->Y is a function, then f(X) is a finite set with #(f(X)) less than or equal to #(X). If in addition f is one to one, then #(f(X)) = #(X)
e) Let X and Y be finite sets. Then cartesian product X x Y is finite and # (X x Y) = #(X) x #(Y)
f) Let X and Y be finite sets. Then the set Y^X is finite and #(Y^X)= #(Y)^#(X)
Provide complete and step by step solution for the question and show calculations and use formulas.