Given three multisets, X, Y, and Z as follows

Question

At LTUC student belong to departments.

• ASAC
• SAE
• Business Administration (BA)

Assume that a set of students in ASAC is A, a set of students in SAE is S, and a set of students in Business Administration is B. A student can belong to two schools at the same time. At graduation ceremonies, You allocated a specific day for each school’s graduation ceremony based on the following rules:

• Sunday will be allocated to students who belong to ASAC and SAE at the same time but not in BA.
• Monday will be allocated to students of BA that don’t belong to any other department
• Tuesday will be allocated to students of ASAC that don’t belong to any other department
• Wednesday will be allocated to student who belong to BA, SAE, and ASAC at the same time.

• Using set notation, write down two algebraic expressions that describe the set of students that will be graduating each day using the three original sets A, S, B, and reduce the expressions to their simplest form.  Given three multisets, X, Y, and Z as follows?

Answer 1

X=A⋂S represents students belonging to both ASAC and SAE.

Y=A⋂B represents those belonging to both ASAC and BA.

Z=S⋂B represents those belonging to both SAE and BA.

X⋂B=Y⋂S=A⋂S⋂B represents those belonging to all three schools.

Consider the following table for 8 LTUC students (or sets of students): a, b, c, d, e, f, g, h.

	ASAC	SAE	BA
a	❌	❌	❌
b	❌	❌	✔️
c	❌	✔️	❌
d	❌	✔️	✔️
e	✔️	❌	❌
f	✔️	❌	✔️
g	✔️	✔️	❌
h	✔️	✔️	✔️

Using the information in this table:

A={ e f g h }, S={  c d g h }, B={ b d f h }, X={ g h } but Sunday={ g }, Monday={ b }, Tuesday={ e }, Wednesday={ h }.

¬B={ a c e g }, ¬A={ a b c d }, ¬S={ a b e f }

Sunday=X⋂¬B or A⋂S⋂¬B, Monday=B⋂¬A⋂¬S, Tuesday=A⋂¬S⋂¬B, Wednesday=A⋂S⋂B.

CHECK

Sunday={ g h } ⋂ { a c e g }={ g }✔️; Monday={ b d f h } ⋂ { a b c d } ⋂ { a b e f }={ b }✔️; Tuesday={ e f g h } ⋂ { a b e f } ⋂ { a c e g }={ e }✔️, Wednesday={ e f g h } ⋂ { c d g h } ⋂ { b d f h }={ h }✔️

However, the question states that a student can belong to two groups or schools (implying not three), so student h could not exist and Wednesday={ } the empty group. This affects all other sets containing h.