2. (15 pts XC) We saw in class that for sets A and B, the cardinality of their u

Question

2. (15 pts XC) We saw in class that for sets A and B, the cardinality of their union is given by: Prove the following analogous rule for the union of three sets: Hint: Start by treating (BUC) as one set and then apply the two-set rule given above, along with set identities from the table in my lecture notes 6/18 (also in the textbook p.130). You may use the two-set rule as a given and it can help to draw a Venn diagram 3. (10 pts) Prove that if A, B, and C are sets, then AnBnC-AuBUC by showing each side is a subset of the other side

Explanation / Answer

P|A U B U C| =

let's split above equation into two parts such as- P|(A U B) U C|

now, by applying cardinality of union -> P|A U B|= P|A|+P|B|

where ,above equation is similar as P|A| = P|AUB| and P|B| = |C|

so we can write it as-

                   = P|A U B| + P|C| - P|(A U B)^C|

again applying cordinality on P|AUB| -
                  = P|A| + P|B| - P|A^B| + P|C| - P|(A^C) U(B^C)|
                  = P|A| + P|B| - P|A^B| + P|C| - [ P|A^C| + P|B^C| - P|A^C|^ |B^C| ]
                  = P|A| + P|B)|+ P|C| - P|A^B| - P|A^C| - P|B^C| + P|A^B^C|

here ^ is intersection sign.

please check the equation as it is undestandable now,

let me know if you didnt get any step.

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