Final answer:
If x is not an element of b and a is a subset of b, then x cannot be an element of a because every element of a subset must also be an element of the set it is a subset of.
Step-by-step explanation:
To prove that if x is not an element of b and a is a subset of b, then x is not an element of a, we can apply the definition of a subset. A set A is considered a subset of a set B if every element of A is also an element of B. Therefore, if x is not an element of B, x cannot possibly be an element of A because if it were, B would have to contain x as well, contradicting the given that x is not in B.
This logical deduction is based on the relationship between sets that if x is an element of A, and A is a subset of B, then x must also be an element of B. It follows that the relationship of x not being in B automatically means x cannot be in A.
In order to prove that if x is not an element of b and a is a subset of b, then x is not an element of a, we can use a proof by contradiction.
Assume that x is an element of a. Since a is a subset of b, this would mean that x is an element of b as well. However, this contradicts our initial assumption that x is not an element of b. Therefore, our assumption was incorrect and we can conclude that if x is not an element of b and a is a subset of b, then x is not an element of a.
For example, let's say we have a set b = {1, 2, 3} and a = {2}. If x = 4, x is not an element of b but a is a subset of b. Therefore, x is not an element of a.