"is smaller than" is a transitive relation
a relation R is said to be transitive relation if and only if
"(x,y) , (y,z) belongs to R then (x,z) also belongs to R "
let x y z be the three eggs in in the given set of eggs if if x is smaller than y and Y is smaller than Z then it implies that X is smaller than Z and hence it is a transitive relation it is not symmetric and reflexive relations and there by it is not a equivalence relation

Transitivity of "is smaller than" over the sets of eggs in a box

Transitivity is an important property of relations that states that if a relation R holds between A and B, and between B and C, then it must also hold between A and C. In the case of "is smaller than" over the sets of eggs in a box, we can ask whether this relation is transitive or not.

Definition of "is smaller than" over the sets of eggs in a box

The relation "is smaller than" over the sets of eggs in a box is defined as follows: Given two sets of eggs, A and B, we say that A is smaller than B if and only if the number of eggs in A is strictly less than the number of eggs in B.

Is "is smaller than" transitive over the sets of eggs in a box?

Yes, the relation "is smaller than" over the sets of eggs in a box is transitive. To see why, suppose we have three sets of eggs, A, B, and C, such that A is smaller than B and B is smaller than C. Then, by definition, we have:

- The number of eggs in A is strictly less than the number of eggs in B.
- The number of eggs in B is strictly less than the number of eggs in C.

Therefore, we can conclude that:

- The number of eggs in A is strictly less than the number of eggs in C.

This means that A is smaller than C, which is exactly what we need to show that "is smaller than" is transitive over the sets of eggs in a box.

Conclusion

In summary, the relation "is smaller than" over the sets of eggs in a box is transitive, which means that if A is smaller than B and B is smaller than C, then A is smaller than C. This property is important for many mathematical and logical applications, and it helps us reason about the relationships between different sets of objects.