Answer to Question #306671 in Discrete Mathematics for ibrahim

Question

(S∪T)∘R= (S∘R)∪(T∘R).

Answer 1

Solution

For the three sets, $A, B$ , and $C$ , We define

$R$ a relation from $A$ to $A$ B.

$S$ and $T$ are two relations from $B$ to $C$

 

Now for both, $(S ∪ T) ∘ R$ and $(S ∘ R) ∪ (T ∘ R)$

$(S ∪ T) ∘ R$ is the subset of $A × C$

And $(S ∘ R) ∪ (T ∘ R)$ is the subset of $A × C$

Therefore, for any arbitrary element $(u, v) ∈ (S ∪ T) ∘ R$

Then there is $v ∈ B$ , such that $(u, v) ∈ R$ and $(u, v) ∈ (S ∪ T)$

Therefore, $(u, v) ∈ S$ and $(u, v) ∈ T$

Hence, $(u, v) ∈ (S ∪ T)$

Similarly, we can prove, $(u, v) ∈ (T ◦ R)$

Therefore, We can write

$(S ∪ T) ∘ R = (S ∘ R) ∪ (T ∘ R)$

Hence Proved