1. Let's first define the terms: 2. A **subjective** function is usually called **surjective** in mathematics. It means every element in the output set (codomain) has at least one element in the input set (domain) that maps to it. In other words, the function covers the entire output space. 3. An **injective** function means every element in the input set maps to a **unique** element in the output set. No two distinct inputs map to the same output. 4. A **bijective** function means the function is both injective **and** surjective. This means every element in the domain maps to a unique element in the codomain, and all codomain elements are covered. 5. To summarize: - Surjective (subjective): Covers all outputs. - Injective: Each input maps to a different output. - Bijective: One-to-one and onto; perfect pairing between domain and codomain. 6. Example: - Consider $f : \{1,2,3\} \to \{a,b\}$ defined by $f(1)=a$, $f(2)=a$, $f(3)=b$. This function is surjective but not injective (because $1$ and $2$ map to the same). - Consider $f : \{1,2,3\} \to \{a,b,c\}$ with $f(1)=a$, $f(2)=b$, $f(3)=c$. This is bijective. 7. I hope this clarifies the difference between subjective (surjective), injective, and bijective functions!