Final answer:
To demonstrate that the union of two finite sets is finite, we consider that a finite set has a countable number of elements. The union set will have at most the sum of the number of elements in both sets, and since the sum of two finite numbers is finite, the union set is finite as well.
Step-by-step explanation:
To show that if a and b are finite sets, then a ∪ b is a finite set, we can consider the definition of finite sets and the union of sets.
A finite set is one that has a limited number of elements, meaning we can count them, and there will be a last element. If set a has m elements and set b has n elements, then the maximum number of elements that the set a ∪ b can have is m + n. This is because the union of two sets includes all the distinct elements from both sets, without repetition. Even if there is some overlap of elements in sets a and b, the union will not count duplicates, hence the actual number of distinct elements in a ∪ b is less than or equal to m + n.
Since both m and n are finite numbers, their sum is also finite. Hence, the union a ∪ b is also a finite set. This explanation illustrates the basic principle of set theory that the union of finite sets is always finite.