Finite vs infinite – choose the finite set: Which of the following sets is finite?

Introduction / Context:
Determining finiteness reduces to counting solutions or assessing whether a definition yields unboundedly many elements. Polynomial equations over integers often yield a finite number of solutions; arithmetic progressions over naturals and geometric collections of lines are typically infinite.

Given Data / Assumptions:

• Quadratic equation over Z
• Even naturals
• All lines through a given point
• All integers strictly greater than −5

Concept / Approach:
Solve the quadratic x^2 − 2x − 3 = 0 to get integer roots; evaluate the other definitions for enumerability without bound.

Step-by-Step Solution:
Factor: (x − 3)(x + 1) = 0 ⇒ x = 3 or x = −1 ⇒ 2 solutions ⇒ finiteEven naturals: {2,4,6,…} ⇒ infiniteLines through a point: infinitely many directions ⇒ infiniteIntegers x > −5: {−4, −3, …} without upper bound ⇒ infinite

Verification / Alternative check:
Use the degree of the polynomial to bound solutions; degree 2 has at most two roots.

Why Other Options Are Wrong:
They each specify sets with infinitely many members by construction (unbounded sequences or uncountable directions serialized as lines).

Common Pitfalls:
Overlooking that “even naturals” and “integers > −5” extend indefinitely; or thinking lines through a point are few—they are limitless.

Final Answer:
I = {x : x ∈ Z and x^2 − 2x − 3 = 0}