# subset and powerset proof by contra

• May 7th 2009, 06:02 PM
pberardi
subset and powerset proof by contra
Where is the contradiction? Also can this be proven directly?

Prove that if x and y are sets such that P(x) is a subset of P(y) then x is a subset of y.

Assume x is not a subset of y
Then there exists an element z in x which is not in y
z is in P(x)

I am stuck can someone show me the contradiction?
• May 7th 2009, 06:26 PM
TheAbstractionist
Quote:

Originally Posted by pberardi
Where is the contradiction? Also can this be proven directly?

Prove that if x and y are sets such that P(x) is a subset of P(y) then x is a subset of y.

Slight correction: $\color{red}\{z\}$ is in $P(x).$
Well, you are given $P(x)$ is a subset of $P(y).$ $\therefore\ \{z\}\in P(x)\ \implies\ \{z\}\in P(y)\ \implies\ z\in y.$ There is your contradiction.