Sets P1,P2,S1,S2 | (P2 intersection S1) C P1,(P1 intersection S2) C P2 and (S1 intersection S2) C (P1 U P2),Demonstrate that (S1 intersection S2) C...

Friday, September 6, 2013

Sets P1,P2,S1,S2 | (P2 intersection S1) C P1,(P1 intersection S2) C P2 and (S1 intersection S2) C (P1 U P2),Demonstrate that (S1 intersection S2) C...

We are given sets $\displaystyle{P}_{{1}},{P}_{{2}},{S}_{{2}},{S}_{{2}}$ and know
that:

1. $\displaystyle{P}_{{2}}\cap{S}_{{1}}\subset$
P_1

2. $\displaystyle{P}_{{1}}\cap{S}_{{2}}\subset$
P_2

3. $\displaystyle{S}_{{1}}\cap{S}_{{2}}\subset{P}_{{1}}\cup$
P_2

We must show that $\displaystyle{S}_{{1}}\cap{S}_{{2}}\subset{P}{1}\cap$
P_2

(In English: show every everything in both S1 and S2
is also in both P1 and
P2)

Proof:

Let $\displaystyle{x}$ be an
element of $\displaystyle{S}_{{1}}\cap{S}_{{2}}$ . This implies that $\displaystyle{x}\in{S}_{{1}}$ and $\displaystyle{x}\in{S}_{{2}}$ . By (3), we also
know that either $\displaystyle{x}\in{P}_{{1}}$ or $\displaystyle{x}\in{P}_{{2}}$ .

Case one:
suppose $\displaystyle{x}\in{P}_{{1}}$ : By (2), we see $\displaystyle{x}\in{S}_{{2}}$ and $\displaystyle{x}\in{P}_{{1}}$ and by (2) $\displaystyle{P}_{{1}}\cap{S}_{{2}}$
subset P_2 $\displaystyle,\therefore$ x in P_2 $\displaystyle.$

Case two: suppose $\displaystyle{x}$
in P_2 $\displaystyle.{\sin{{c}}}{e}$ x in S_1 $\displaystyle{\quad\text{and}\quad}$ P_2 nn S_1 subset P_1 $\displaystyle,{i}{t}{f{{o}}}{l}{l}{o}{w}{s}{t}{\hat{}}$ x in P_1
.

Since each of these cases shows x must exist in both P1
and P2, it follows that $\displaystyle{S}_{{1}}\cap{S}_{{2}}\subset{P}_{{1}}\cap{P}_{{2}}$

Help Notes

Friday, September 6, 2013