Prove that for every set S, Ø \subseteq( S.
i need to Use vacuous proof.
i know that vacuous proof is when the hypothesis is always False.
but for this i find it very difficult please can you help me prove this? thanks.
a simple web search for
prove null set is a subset of every set
will provide many useful proofs.
As a rule, you might try a web search before posting questions that can be answered with a paragraph, rather than a mathematical solution.
To prove that for every set S, the empty set Ø is a subset of S, we can use a vacuous proof.
A vacuous proof is used when the hypothesis is always false, which means that there are no elements to consider in the hypothesis. In this case, the hypothesis is the statement "Ø is a subset of S" which means we assume that the empty set is not a subset of S.
To prove that Ø is a subset of S, we need to show that every element in the empty set is also an element of S. However, since the empty set has no elements, there is nothing to consider or prove. Therefore, the statement "Ø is a subset of S" is true because there are no counterexamples.
In other words, any claim about the empty set is considered true by default because it does not have any elements to disprove the claim. So, the statement "Ø is a subset of S" is true for any set S.
In a formal proof, you can state it as follows:
1. Assume Ø is not a subset of S (for contradiction).
2. But the empty set has no elements.
3. Therefore, there are no elements in Ø that are not in S.
4. Hence, every element in Ø is also in S.
5. This contradicts the assumption that Ø is not a subset of S.
6. Therefore, Ø must be a subset of S.