Question

If R is a relation from a set A to set B and S is a relation from B to a set C, then the relation SoR
A. Is from A to C
B. Is from C to A
C. Does not exist
D. None of these

Answer 1

Hint: A relation between two sets is a collection of ordered pairs containing one object from each set. Use the fact that if P is a relation from set E to set F and Q is a relation from set F to set G then the relation QoP contains the ordered pairs $(e,g), \subseteq E \times G$

Complete step by step Solution:
We know that R is a relation from a set A to a set B and S is a relation from set B to a set C. The composition denoted by SoR is the relation from set A to set C and it contains the ordered pairs (a,c) where $a \in A$ and $c \in C$ for which there exist $b \in B$ such that $(a,b) \in R$ and $(b,c) \in S$.
Mathematically,
$R = \{ (a,b), \subseteq A \times B\} $
$S = \{ (b,c), \subseteq B \times C\} $
$SoR = \{ (a,c), \subseteq A \times C|\,\forall \,b \subseteq B:(a,b) \subseteq R,\,(b,c) \subseteq S\} $
Therefore, SoR is from set A to set C.

Hence, the correct option is (A).

Note: If A and B are sets, the Cartesian product of A and B is the set $A \times B = \{ (a,b):(a \in A)\,{\text{and (b}} \in B)\} $. A relation R from a set A to a set B is, therefore, a subset of $A \times B$, i.e., $R \subseteq A \times B$