Question

which venn diagram represents the set relationship $a \subset b$?

Explanation:

The symbol \( A \subset B \) means that set \( A \) is a proper subset of set \( B \), which implies that every element of \( A \) is also an element of \( B \), and \( A \) is not equal to \( B \) (so there are elements in \( B \) that are not in \( A \)).

To represent this in a Venn diagram, the circle representing \( A \) should be entirely inside the circle representing \( B \), with \( B \) being larger (or at least containing \( A \) with some extra area).

Let's analyze each diagram:

1. First diagram: \( A \) and \( B \) are separate circles. This represents disjoint sets (no elements in common), so not \( A \subset B \).
2. Second diagram: \( A \) and \( B \) overlap (intersection), but neither is entirely inside the other. This represents sets with some common elements and some unique to each, so not \( A \subset B \).
3. Third diagram: \( B \) is inside \( A \). This would represent \( B \subset A \), not \( A \subset B \).
4. Fourth diagram: \( A \) is entirely inside \( B \), and \( B \) has an area outside \( A \). This matches the definition of \( A \subset B \).

Answer:

The fourth Venn diagram (the one where the circle labeled A is entirely inside the circle labeled B, and B has a region outside A) represents \( A \subset B \).