Theory:

Let $B,C,D$ be any sets, $R,S$ be any relation.

a$)$ Prove: If $B$CsubeD and $CD=\frac{O}{?},$ then $C=\frac{O}{?}.$

b$)$ Prove: If $P(B)subeP(CD),$ then BsubeC.

c$)$ Disprove: If $R$ on $B$ is transitive and $S$ on $C$ is transitive, then $RS$ on $BC$ is transitive.

Note: All proofs must be done using the sets definitions given in the notes.

A Venn Diagram is not a proof and will not earn you part marks.Application:

Survey a bunch of your classmates who are enrolled in classes this term. Do not keep track of how many you survey.

Make each participant selects all that are applicable:

i$)$ I am enrolled in a faculty of science class.

ii$)$ I am enrolled in a faculty of arts class.

iii$)$ I am enrolled in a class that is in another faculty.

a$)$ Give the formula for the principle of inclusion$-$exclusion for two sets. For each pair of sets, state the values for each set and intersection.

b$)$ Calculate the unions for the sets.

c$)$ Repeat parts a$)$ and b$)$ but now with the formula for three sets. Also, draw the Venn diagram.