Given the relation \( R = \{(1, 2), (2, 3), (3, 4), (4, 5)\} \), determine if it is bijective.
Answer
To check if \( R \) is surjective, we need to see if every element of the codomain is mapped to by an element in the domain. In this case, every element of the codomain (\{2, 3, 4, 5\}) is mapped to by an element in the domain (\{1, 2, 3, 4\}), so \( R \) is surjective.
Step 1 ：A relation \( R \) is bijective if it is both injective (one-to-one) and surjective (onto).
Step 2 ：To check if \( R \) is injective, we need to see if each element of the domain maps to a unique element in the codomain. In this case, no element of the domain (\{1, 2, 3, 4\}) maps to more than one element in the codomain (\{2, 3, 4, 5\}), so \( R \) is injective.
Step 3 ：To check if \( R \) is surjective, we need to see if every element of the codomain is mapped to by an element in the domain. In this case, every element of the codomain (\{2, 3, 4, 5\}) is mapped to by an element in the domain (\{1, 2, 3, 4\}), so \( R \) is surjective.
