Question

Decide whether the following arguments are valid or not. State the Rule of Inference or fallacy used. i. Let A,B and C be three sets. Prove by contradiction that if A∩B⊆C and x∈B, then x∈/A−C.

Answer 1

The given argument that if A intersect B is a subset of C and x is in B, then x is not in A-C is valid based on the proof by contradiction. The approach assumes the conclusion is false and shows this leads to a contradiction with the premises thus confirming the argument's validity.

Step-by-step explanation:

The student asked about the validity of an argument involving sets and elements. The argument states that if A∩B ⊆ C and x ∈ B, then x ∉ A-C. To determine whether this argument is valid or not, we employ a proof by contradiction.

Step-by-Step Explanation

1. Assume the conclusion is false, which means x ∈ A-C. This implies x ∈ A and x ∉ C.
2. Given the first premise A∩B ⊆ C, if x ∈ A and x ∈ B, it would follow that x ∈ C, contradicting our assumption that x ∉ C.
3. Therefore, our initial assumption must be wrong, and we conclude that if A∩B ⊆ C and x ∈ B, then it must be the case that x ∉ A-C.

This argument uses a proof by contradiction, where the negation of the conclusion leads to a contradiction with the premises. Thus, the argument is valid as the form ensures that if the premises are true, the conclusion must also be true.