Mathematics: Functions and POSet Concepts

What is the inverse of the function defined by the simple cipher function $f(x) = 3x - 4$?

For the power set of the set $X = {a, b, c, d}$, how many elements does the power set have?

If 60 students are surveyed, with 40 liking math and 36 liking science, how many students like either math or science?

Which statement correctly defines the relation $R = {(a, b) \mid a \text{ divides } b}$ on the set $A = {1, 2, 3, 4, 6, 8, 12, 24}$?

What is the total family income if a salesperson and his son both sell Rupees 1,50,000 worth of merchandise?

Simple Cipher Function

A simple cipher codes a number using the function f(x) = 3x - 4
Find the inverse function
Determine if the inverse is a function
Verify symmetrical property about the line y = x

POSet Definition

Define POSet (Partially Ordered Set)
Show that (X, ≤) is a POSet
X = {a, b, c} (example set)
Draw the Hasse diagram for the power set of X

More POSet Examples

Define POSet
Show (X, ≤) is a POSet, where X = {a, b, c, d}
Draw Hasse diagram for the power set of X

Relation as POSet

Let A = {2, 3, 6, 12, 24, 36}
Relation R = {(a, b) | a divides b}
Show (A, R) is a POSet
Draw the Hasse diagram

Another Relation as POSet

Let A = {1, 2, 3, 4, 6, 8, 12, 24}
Relation R = {(a, b) | a divides b}
Show (A, R) is a POSet
Draw the Hasse diagram

Student Survey

60 students total
40 like math
36 like science
24 like both math and science
Calculate students who like:
- Math only
- Science only
- Either math or science
- Neither math nor science

Salesperson Earnings

Salesperson's annual earnings: A(x) = 30,000 + 0.04x
Son's annual earnings: S(x) = 25,000 + 0.05x
x = rupee value of merchandise
Find (A + S)(x)
Calculate total family income if they each sell ₹1,50,00,000 worth of merchandise

Equivalence Relation

Define Equivalence relation
Show that R = {(x, y) | x² = y²} is an equivalence relation on S = ℝ

Function Composition

f(x) = x²
g(x) = 3x
h(x) = x²
Prove (fog) o h = f o (go h)

Boolean Functions

Write Boolean function for Boolean variables {0, 1}
F(x, y) = xy

This quiz covers important concepts in mathematics, including simple cipher functions and their inverses, as well as definitions and examples of partially ordered sets (POSets). Additionally, it includes exercises on relations as POSet and a survey of student preferences in mathematics and science. Test your understanding of these topics with this comprehensive quiz.