What is [math: T∪N]?

Question

BlackTom AI · Accepted Answer

If one set is contained in another, their union equals the larger set. This key set-theory fact is the basis for answering T ∪ N.

1) Relevant knowledge points and concepts
- Union: A ∪ B = {x | x ∈ A or x ∈ B}.
- Subset: A ⊆ B means every element of A is also an element of B.
- Important identity: If A ⊆ B then A ∪ B = B.

2) Step-by-step reasoning
- Assume T ⊆ N (this is the implicit context needed to make the multiple-choice answer N correct). Under that assumption:
  Step 1: Take any element x ∈ T ∪ N. By definition of union, x ∈ T or x ∈ N.
  Step 2: If x ∈ T, since T ⊆ N we have x ∈ N. If x ∈ N already, then x ∈ N. So in every case x ∈ N. Therefore T ∪ N ⊆ N.
  Step 3: Conversely, take any element y ∈ N. Then y ∈ T ∪ N because union includes all elements of N. So N ⊆ T ∪ N.
  Step 4: From Steps 2 and 3 we obtain equality: T ∪ N = N.

3) Intermediate logical steps shown above: (a) show every element of the union lies in N, (b) show every element of N lies in the union, (c) conclude equality.

4) Why the correct answer is correct
- Given T is contained in N, adding T to N via union does not introduce any element outside N. Thus the union is exactly N.

5) Why the other options are incorrect
- T: would be correct only if N ⊆ T (i.e., if T were the larger set), which contradicts the assumption T ⊆ N.
- Z or P: these denote different sets (integers Z, primes P) and are not forced to equal T ∪ N by the union identity.
- None of the above: incorrect because the union identity specifically yields N under the subset relation.

Conclusion: By the subset-union identity and the reasoning above, T ∪ N = N, which matches the provided correct answer N.

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