Question

Solve the following two " union " type questions:
(a) How many bit strings of length 8 either begin with 2 0s or end with 1 1s? (inclusive or)
(b) Every student in a discrete math class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 36 computer science majors (including joint majors), 29 math majors (including joint majors) and 5 joint majors?

Answer 1

(a) There are a total of $|A|=2^6$ strings that start with two 0's and $|B|=2^7$ strings that ends with 1. We need to find the cardinality of a set $A \cup B$. The Sum Rule gives us $|A \cup B| = |A| + |B|-|A\cap B|$. Set $A \cup B$ contains strings of length 8 that both begin with 00 and end with 1 ($|A \cup B|=2^5$).

Answer: $|A \cup B| = 2^6 + 2^7-2^5=64+128-32=160.$

(b) Let $A$ be a set of computer science majors and $B$ represents maths majors. We need to find the cardinality of set $A\cup B$. We are given $|A|=36, \ |B|=29, \ |A\cap B|=5$. The cardinality of a union of the two following sets is as follows

$|A \cup B| = |A| + |B|-|A\cap B|=36+29-5=60$.

Answer: 60 students.