Question

Suppose \(A \neq \varnothing\). Prove that \(A \times B \subseteq A \times C\) if and only if \(B \subseteq C\).

Short answer

The statement is proven to be true. We've shown that if \(A \times B \subseteq A \times C\) then \(B \subseteq C\), and conversely, if \(B \subseteq C\) then \(A \times B \subseteq A \times C\).

Step-by-step solution

Proof of 'if' case

Assume that \(A \times B \subseteq A \times C\). Pick an arbitrary element \(b\) from set \(B\). Since \(A\) is non-empty, there exists some element \(a\) in \(A\). Therefore, the ordered pair \((a,b)\) is in \(A \times B\). Since we've assumed \(A \times B \subseteq A \times C\), the pair \((a,b)\) must also be in \(A \times C\). But by definition of the cartesian product, there should then exist an element \(c\) in \(C\) such that \((a,c) = (a,b)\). From the property of ordered pairs, it is known that if two ordered pairs are equal, their respective components are also equal. Hence, \(c = b\). Therefore, for every element \(b\) chosen from \(B\), there exists a corresponding element in \(C\). Thus, \(B \subseteq C\).

Proof of 'only if' case

Now take the second case, where \(B \subseteq C\). We need to show that \(A \times B \subseteq A \times C\). Pick an arbitrary element \((a,b)\) from \(A \times B\). From the definition of cartesian product, \(a\) in \(A\) and \(b\) in \(B\). Since we've assumed \(B \subseteq C\), it must also be the case that \(b\) is an element of \(C\). Therefore, the pair \((a,b)\) is in \(A \times C\). Therefore, for every element \((a,b)\) chosen from \(A \times B\), there exists a corresponding element in \(A \times C\). Thus, \(A \times B \subseteq A \times C\).

Key concepts

Set Theory

Set theory is the mathematical science of sets, which are collections of objects that can be clearly defined and distinguished from each other. In the context of the given problem, we're dealing with sets represented by the letters A, B, and C, and their Cartesian products, denoted as A × B and A × C.

A crucial concept in set theory that this problem illustrates is the subset relation, which defines when all elements of one set (say, set B) are also elements of another set (set C). This is notated as B ⊆ C. The problem also introduces the Cartesian product, a set formed by pairs of elements from two given sets, where each pair consists of one element from each set.

Ordered Pair Property

The ordered pair property is fundamental when dealing with Cartesian products. An ordered pair, such as (a, b), has an important feature: it is only equal to another pair, (c, d), if and only if a = c and b = d. This precise definition allows mathematicians to compare paired elements from different sets unambiguously.

In the exercise, when we encounter the ordered pair (a, b) from the set A × B, and we state that A × B ⊆ A × C, we use the ordered pair property to deduce that the second element of the pair, b, must also belong to set C. This, in return, aids in proving the 'if' and the 'only if' parts of the biconditional statement by establishing a clear relation between the elements of set B and set C.

Subset Relation

The subset relation is a way of expressing that all elements of one set are also contained within another set. In the provided exercise, we use the subset relation to describe that set B is a subset of set C, or in other words, B ⊆ C.

The implication of this relation for the Cartesian products A × B and A × C is that if B is indeed a subset of C, each pair from A × B will naturally map onto a unique pair in A × C. This is because the first element of the pair comes from set A, which remains constant, and the second element from set B, which is already contained within set C, fulfilling the condition of B ⊆ C.

Proof Techniques

In mathematics, proof techniques are methods used to verify the truth of propositions. The problem we've analyzed uses a direct proof approach. This involves assuming an initial statement, like B ⊆ C, and logically deducing that another condition follows from it, like A × B ⊆ A × C.

The proof is done in two parts, one for each direction of the biconditional 'if and only if' statement. First, we show that if A × B is a subset of A × C, then B must be a subset of C. Then, we prove that if B is a subset of C, then A × B must be a subset of A × C. These steps require careful logical reasoning and a thorough understanding of how ordered pairs and subset relations work in set theory.