query: Proving the predicate $\forall n \in \mathbb{N} , n \geq 2$ using the Pigeonhole principle
positive_passages.docid: 151935
positive_passages.text: Proving the predicate $\forall n \in \mathbb{N} , n \geq 2$ using the Pigeonhole principle
negative_passages.docid: 139078
negative_passages.text: Infinite partition of $\mathbb N$ by infinite subsets
negative_passages.docid: 45460
negative_passages.text: Proof verification of a number theory problem involving sequences.
negative_passages.docid: 78163
negative_passages.text: Hint for problem on $4 \times 7$-chessboard problem related to pigeonhole principle
negative_passages.docid: 123613
negative_passages.text: Double Complement of a set proof
negative_passages.docid: 1409
negative_passages.text: Proving that the given set has at most $2^{n-1}$ elements
negative_passages.docid: 12857
negative_passages.text: Induction proof: $S$ contains powers of 2 and predecessors implies $S={\bf N}$
negative_passages.docid: 94923
negative_passages.text: Using the Pigeonhole Principle to show that $2$ of any $n+1$ numbers from $\{1,2,\ldots,2n\}$ sum to $2n+1$
negative_passages.docid: 39670
negative_passages.text: Use Pigeonhole to show of any set of $2^{n+1}-1$ positive integers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$.
negative_passages.docid: 48368
negative_passages.text: Prove using Natural Deduction in predicate logic $(\forall x\ P(x))\to Q ⊢ \exists x (P(x)\to Q)$
negative_passages.docid: 74730
negative_passages.text: Combinatorial Proof of ${{n}\choose{k}} {{k}\choose{j}} = {{n}\choose{j}} {{n-j}\choose{k-j}}$
negative_passages.docid: 59885
negative_passages.text: Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence
negative_passages.docid: 61671
negative_passages.text: Discrete Math: Writing Proofs
negative_passages.docid: 130093
negative_passages.text: Prove by Induction : $n < 2^n$
negative_passages.docid: 9651
negative_passages.text: Cardinality of cartesian product of an infinite set with N
negative_passages.docid: 95000
negative_passages.text: Using induction to prove that $2 \mid (n^2 − n)$ for $n\geq 1$
negative_passages.docid: 104309
negative_passages.text: Pigeon-Hole Principle Common Sum