# Proper Subsets and Relations of Sets

• MHB
In summary: And looking at R, we can see that it does indeed have these elements. Therefore, R is reflexive. In summary, a proper subset of S is any subset of S that is not equal to S. And R is reflective as it contains elements (x,x), (y,y), and (z,z).
Q1: Write all proper subsets of S = {1, 2, 3, 4 }.

Q2: Let S = {1,2,5,6 }
Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)

Q3: Let S = {x, y, z } and R is a relation defined on S such that
R={(y,y),(x,z),(z,x),(x,x),(z,z),(x,y),(y,x)}
Show that R is reflneed proper solution

Q1: Write all proper subsets of S = {1, 2, 3, 4 }.

Q2: Let S = {1,2,5,6 }
Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)

Q3: Let S = {x, y, z } and R is a relation defined on S such that
R={(y,y),(x,z),(z,x),(x,x),(z,z),(x,y),(y,x)}
Show that R is reflneed proper solution

Q1:
A proper subset of $S$ is any subset of $S$ that is not equal to $S$.
So we have...
$\emptyset$
$\left\{1\right\}, \left\{2\right\}, \left\{3\right\}, \left\{4\right\}$
$\left\{1,2\right\}, \left\{1,3\right\}, \left\{1,4\right\}, \left\{2,3\right\}, \left\{2,4\right\}, \left\{3,4\right\}$
$\left\{1,2,3\right\}, \left\{2,3,4\right\}, \left\{1,3,4\right\}, \left\{1,2,4\right\}$

Q3:
R is reflective if for every element $s$ of $S$, $(s,s)$ is in $R$.
So what do you need to check?
For R to be reflexive, it must have elements $(x,x)$, $(y,y)$, and $(z,z)$.

## 1. What is a proper subset?

A proper subset is a subset of a given set, where all the elements of the subset are also elements of the original set, but the subset is not equal to the original set. In other words, a proper subset is a smaller version of the original set, with at least one element removed.

## 2. How is a proper subset represented?

A proper subset is represented using the symbol ⊂ (subset of) or ⊊ (proper subset of). For example, if set A is a proper subset of set B, it would be written as A ⊂ B or A ⊊ B.

## 3. What is the difference between a proper subset and an improper subset?

The main difference between a proper subset and an improper subset is that an improper subset can be equal to the original set, while a proper subset cannot. This means that an improper subset can have all the elements of the original set, while a proper subset will always have at least one less element.

## 4. How is a relation between sets defined?

A relation between sets is a rule or connection that describes how the elements of one set are related to the elements of another set. This can be represented in different ways, such as through a table, a diagram, or a formula.

## 5. What are the different types of relations between sets?

There are several types of relations between sets, including one-to-one, one-to-many, many-to-one, and many-to-many. In a one-to-one relation, each element of one set is related to exactly one element of the other set. In a one-to-many relation, each element of one set is related to multiple elements of the other set. In a many-to-one relation, multiple elements of one set are related to one element of the other set. In a many-to-many relation, multiple elements of one set are related to multiple elements of the other set.

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