Question

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) If \(A\) and \(B\) are sets, then \(A \cap(B-A)=\varnothing\).

Short answer

The statement, if A and B are sets, then \(A \cap (B - A) = \varnothing\) is proven by contradiction. We began by assuming the opposite, that there exists an element x in both sets A and \(B - A\), then found a contradiction as no element can exist in B but not in A and exist in A at the same time, leading us to the conclusion that our assumption must be wrong. Thus, if A and B are sets, then \(A \cap (B - A) = \varnothing\).

Step-by-step solution

Understanding terms

In this problem, we have two sets A and B. The expression \(B - A\) is used to denote the set of elements that belong to B but not to A. The intersection of two sets \(A \cap B\) is the set of elements common to both A and B.

Assume the opposite

For a proof by contradiction, an assumption is made that the opposite of what is trying to be proven is true. So, we assume that there exists an element x that belongs to both A and also to the set \(B - A\). Meaning that \(x \in A \cap (B - A)\).

Find the contradiction

If the element x belongs to both A and the set \(B - A\), this means that x must belong to set A (since x belongs to \(A \cap (B - A)\)) and x must also belong to B but not A (since x belongs to \(B - A\)) at the same time, which is contradictory.

Step 4:Conclude the proof

Since we've found a contradiction when we assumed there existed an element in both A and \(B - A\), our assumption must be incorrect. Therefore, it's proven that if A and B are sets, then \(A \cap (B - A) = \varnothing\). This completes the proof.

Key concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with the collection of objects, known as sets. In essence, sets are groups of elements that can be anything: numbers, symbols, points in space, etc. Each element in a set is unique; no duplicates are allowed. Sets are typically denoted by capital letters, such as A, B, or C, and can be described by listing their elements or by a characteristic property.

For instance, a set A might be described as \( A = \{x \mid x \text{ is an even number greater than 3} \} \), which reads 'A is the set of all elements \( x \) such that \( x \) is an even number greater than 3'. Understanding set theory is crucial for studying more advanced mathematical concepts, and it provides the foundational language for describing and manipulating collections of objects.

Intersection of Sets

The intersection of sets refers to the set containing all elements that are common to both sets. Symbolically, the intersection of two sets A and B is denoted as \( A \cap B \).

For example, if \( A = \{1, 2, 3, 4\} \) and \( B = \{3, 4, 5, 6\} \), the intersection \( A \cap B \) would be \( \{3, 4\} \), as these are the elements present in both sets. When the intersection of two sets is empty, meaning they have no elements in common, it is denoted as \( \varnothing \) or \( \emptyset \) and called an empty set or null set. An empty set is a clear indication that two sets do not share any common elements.

Complement of Sets

The complement of a set A, often denoted by \( A^c \) or \( \overline{A} \) when considering a universal set U, consists of all elements that are in U but not in A. If no universal set is mentioned, it is typically assumed based on the context.

When discussing the difference between two sets, such as \( B - A \) (also written as \( B \setminus A \) or \( B \backslash A \) ), we are essentially referring to the set of elements that belong to set B but not to set A. In the context of the given textbook problem, \( B - A \) would contain all the elements that are uniquely in B when considering elements from set A as well.

Logical Contradiction

Logical contradiction occurs when two conditions cannot possibly coexist because they are in direct opposition to each other. This is a key element in proof by contradiction, a mathematical technique where the assumption of the opposite of what is to be proven is shown to lead to a contradiction. If assuming the opposite yields an impossible situation, then the original statement must be true.

In the context of our set theory exercise, finding an element that belongs to both A and \( B - A \) is an immediate contradiction because by definition, \( B - A \) is the set of all elements in B that are not in A. Thus, it is logically impossible for the same element to be in both A and \( B - A \) at the same time.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of a mathematical statement beyond any doubt. They use a series of logical deductions based on axioms, definitions, and previously established theorems. The structure of a proof can vary: some are direct, some use the contrapositive, and some, like the proof by contradiction in this exercise, assume the opposite to show that a statement must be true.

Proof by contradiction is particularly powerful because it can sometimes simplify complex problems by focusing on the impossibility of the alternative. When students study a proof by contradiction, they learn how to think critically about the implications of their assumptions and learn to articulate why those assumptions cannot hold true.