For any set X construct an injection f X rightarrow PXSoluti

For any set X, construct an injection f: X rightarrow P(X).

Solution

assume X to be {1,2,3} then P(x) will have {{1},{1,2},{1,2,3},{1,3},{3,2},{3,2,1},etc}}

associate to each element x of X an element (just one) of P(X). The elements of P(X) are ALL subsets of X, so you should associate to each x a subset of X (which is considered as one element of P(X)). For example you can associate to each x of X the empty set which is an element of P(X). But of course this is not injective; it is a consatante map. i suggest for each x to associate le subset of X which contains only the element x, that is {x}. Why this is an injection? Answer: Assume that there is x and y in X such that f(x)=f(y) then {x}={y} (because by definition f(x)={x} and f(y)={y}, remember we have said that for EACH x we associate {x} so to y we associate {y}) Now because the set {x}={y} then each element of {x} is also an element of {y} so, the unique element x of {x} is an element of {y} which contains only y, so there is no choice x has to be y, so, x=y. The injection is proved.