Question

Prove that if $A\subseteq B,$ then $R[A]\subseteq R[B].$ Disprove the converse by a counterexample.

Short answer

Yes, if $A\subseteq B$, then $R[A]\subseteq R[B]$. The converse is false, as demonstrated with a counterexample.

Step-by-step solution

Understand the Sets

We need to understand what it means for one set to be a subset of another. If $A\subseteq B$, it implies that every element of set $A$ is also an element of set $B$.

Apply the Function

Assume a function $R$ that maps elements from one set to another. For an element $a$ in set $A$, $R[a]$ maps the element $a$ to its image in the range. If $A\subseteq B$, then every element $a$ of $A$ is part of $B$. Thus, $R[a]\in R[B]$ as well, because $A^{\prime }s$ elements are also in $B$.

Prove R[A] is Subset of R[B]

Since every element $a$ in $A$ is also in $B$, the image of $a$, i.e., $R[a]$, is an element of both $R[A]$ and $R[B]$. Thus, we can conclude $R[A]\subseteq R[B]$.

Disprove the Converse with Counterexample

For the converse, assume that $R[A]\subseteq R[B]$ implies $A\subseteq B$. Consider $A=\{1,2\}$ and $B=\{2,3\}$, and let $R[x]=x^2$. Then $R[A]=\{1,4\}$ and $R[B]=\{4,9\}$. We see $R[A]ot\subseteq R[B]$ even though $4\in R[B]$, and $Aot\subseteq B$ because $1otinB$. This shows that even if $R[A]\subseteq R[B]$, $A\subseteq B$ does not necessarily hold.

Key concepts

Subset

A subset is a fundamental concept in set theory. When we say that "A is a subset of B," denoted as $A\subseteq B$, it means that every element of the set $A$ is also an element of the set $B$.
For example, consider the set $A=\{1,2\}$ and the set $B=\{1,2,3\}$. Here, $A$ is a subset of $B$, because each element of $A$ (which are 1 and 2) can also be found in $B$.

• If a set $A$ has no elements, or is an empty set $\mathrm{\varnothing }$, it is also considered a subset of any set $B$, because there is no element in $A$ that is not in $B$.
• A set $A$ can be a subset of itself, which is known as the improper subset.

Understanding subsets is crucial for mapping relationships between sets and proving related mathematical statements.

Function

In mathematics, a function maps elements from one set, called the domain, to another set, called the range.
A function is a rule that assigns each input exactly one output.
If we have a function $R$ and a set $A$, the image of $A$ under $R$ is $R[A]$. This image consists of all outputs that result from applying the function rule to inputs from $A$.

• For example, if $R[x]=x^2$ and $A=\{1,2\}$, then $R[A]=\{1^2,2^2\}=\{1,4\}$.
• The domain is the set of inputs for which the function is defined.
• The range is the set of all possible outputs.

Functions are essential tools in mathematics for relating different sets, especially when seeking to understand complex relationships.

Counterexample

A counterexample is a specific case that disproves a general statement or proposition.
In proving or disproving mathematical statements, a single counterexample is sufficient to show that a proposition does not hold universally.
When we assume the converse that if $R[A]\subseteq R[B]$, then $A\subseteq B$, we disprove it using a counterexample.

• Consider $A=\{1,2\}$, $B=\{2,3\}$, and $R[x]=x^2$. We find that $R[A]=\{1,4\}$, and $R[B]=\{4,9\}$.
• Here, $R[A]subseteqR[B]$ because 1 is not in $R[B]$, even though $4$ is. Similarly, $AsubseteqB$, as 1 is not in $B$.

This counterexample demonstrates that the converse does not hold true, highlighting how detailed examination of examples is necessary in mathematics.

Range

The range of a function is the set of all possible outputs. It is a crucial aspect that defines the image of elements from the function's domain.
When mapping elements using a function, attention to the range helps determine the reach of the function outputs.
For instance, with the function $R[x]=x^2$ applied to sets $A=\{1,2\}$ and $B=\{2,3\}$, we calculate:

• $R[A]=\{1,4\}$
• $R[B]=\{4,9\}$

Noting the range helps us understand which elements are produced as outputs across different inputs, thus portraying the function's effectivity visually and practically.
By understanding the range, we assess the overall behavior and limitations of the functions applied in set theory problems.