Problem # $3-$ Challenge Set $(22$ pts$)$

This problem is going to explore programming languages through set theory. You don$$t need to know

anything about different programming languages to answer these questions, but you may learn some new

things about programming languages by applying the set theory to this problem. First, let$$s define a few sets:

P $=\{$Python$,$ C$++,$ Java, C$,$ C#$,$ Javascript, Visual Basic, Go$,$ SQL$,$ Fortran$\}$

H $=\{$C$,$ C$++,$ Fortran, Lisp, Basic$\}$

G $=\{$Python$,$ Java, Javascript, Visual Basic, Go$,$ SQL$,$ Scheme, Haskell, Lisp, R$,$ MATLAB, Smalltalk$\}$

I $=\{$R$,$ MATLAB, Lisp, Python, Javascript$\}$

The set P is the top ten most popular programming languages right now $($cite$).$ The set H is the programming

languages that are over $35$ years old $($cite$).$ The set G is the programming languages which have a garbage

collector $($cite$).$ A garbage collector means that the programming langauge automatically manages memory

$($you will learn about this in $141,211,$ and $251).$ The set I is the programming languages which can

be categorized as interpreted languages $($cite$).$ If you took $111,$ you know that Python is an interpreted

language $($which means it does not require you to compile it before running it$).$

a Draw the Venn Diagram for P$,$ H$,$ and I. Assume the universal set U is made up of all programming

languages mentioned above in any set above $($P$,$ H$,$ G$,$ and I$).$

b In set notation, represent the area in your Venn Diagram which is equal to $.$ Your answer should be

in terms of sets P$,$ H$,$ and I and set operations $(,,,,$ X$,$ etc.$).$ Explain in English what this

set represents.

CS $151$: Mathematical Foundations of Computing $$ Fall $20242$

Katok & Reckinger Homework#$3$ Due: October $20$th$,2024$

c Write out the contents of the set $($G$$P $$I$)$G $$ P $$ I. Describe in English what this set represents.

Look at the follow sets:

P $$ G

P $$ H $$ I

P $$ H $$ G

H $$ I

P $$ I

P $$ H $$ G

G $$ P $$ H

P $$ H $$ G

P $$ H

d Which of the sets have a cardinality of $0?$

e For any sets with cardinality $0,$ can you find a programming language not listed in our Universal set

that would make these sets non$-$zero?

f Which of the sets listed have cardinality of $1?$

g Write out the sets listed that have cardinality greater than or equal to $5.$