Question

Let $R$ be a relation from a set $A$ to a set $B$, then:
A) $R = A \cup B$
B) $R = A \cap B$
C) $R \subseteq A \times B$
D) $R \subseteq B \times A$

Answer 1

Here we will be using the property of sets and Venn diagrams which states that if any relation is from a set $A$ to set $B$ then it is a subset of $A \times B$ where the subset symbol denotes as $\subseteq$.

Complete step-by-step solution:
Step 1: Let set A = \left\\{ {1,2} \right\\} and set B = \left\\{ a \right\\}, $A \times B$ will be equals to as below:
\Rightarrow A \times B = \left\\{ {\left( {1,a} \right),\left( {2,a} \right)} \right\\}
But as given in the question that
$R$ is a relation from a set $A$ to a set $B$, so we can write the relation as below:
$\Rightarrow R = A \times B$
Which also means that relation $R$ is a subset of $A \times B$.

$\because$ The relation between $R$ and $A \times B$ is $R \subseteq A \times B$.

Note: Students needs to remember some important points of sets and relations as mentioned below:
If a set A = \left\\{ {1,2,3,4} \right\\} and set B = \left\\{ {0,1,2,3,4} \right\\}, then we can say that set $A$ is a subset of the set
$B$ because all the elements $A$ belong to $B$, so we can write as $A \subseteq B$. But the set $B$ is not a subset of $A$ because $0 \notin A$. So, we can write as $B \not\subset A$
Any set $S$ is a subset of itself because every element $S$ is an element of $S$.
An empty set is always a subset of every set $S$ because every element of an empty set is an element of $S$. An empty set is denoted by the symbol $\phi$ and it means that there are no elements in it.
Suppose $S$ is a finite set( which means that we can list all of its elements ), then the symbol $\left| S \right|$ denotes the number of elements in that set.