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

Hint: 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.