Final answer:
An explicit bijection between the intervals [1,2] and [3,5] can be determined using a linear function. The function is f(x) = 2x + 1, as it maps the endpoints of [1,2] to the endpoints of [3,5], proving that the two intervals have the same cardinality.
Step-by-step explanation:
To establish a bijection between the intervals [1,2] and [3,5], we need to find a function f that pairs every element x in [1,2] with a unique element f(x) in [3,5], and vice versa. One way to construct such a bijection is to find a linear function that maps the endpoints of the interval [1,2] to the endpoints of the interval [3,5]. To do this, we can use the formula for a line f(x) = a × x + b, where a is the slope and b is the y-intercept.
Since we want f(1) = 3 and f(2) = 5, we have two equations: a + b = 3 and 2a + b = 5. Solving these simultaneously gives us a = 2 and b = 1. Therefore, the explicit bijection is f(x) = 2x + 1. This function is a bijection as it is both injective (one-to-one) and surjective (onto), which shows that the intervals [1,2] and [3,5] have the same cardinality.