Final answer:
To prove that if set a is a subset of set b and set a is a subset of set c, then set a is a subset of the intersection b∩c, one can apply the definitions of a subset and intersection. All elements of a are in both b and c, thus a is a subset of b∩c, proving the statement.
Step-by-step explanation:
To prove that if a is a subset of b and a is a subset of c, then a is a subset of b intersection c (b∩c), let's use the definition of a subset and the definition of the intersection of two sets.
By definition, if a is a subset of b, every element of a is also an element of b. Similarly, if a is a subset of c, every element of a is also an element of c. The intersection b∩c consists of all elements that are both in b and in c. Therefore, since every element of a is in both b and c, by extension, every element of a must be in b∩c. Hence, a is a subset of b∩c, which completes the proof.