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and P0(x1,x2,…,xn) = 1 = Q0(x1,x2,…,xn). Show that for any d > 0:
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2. Let α ∈ R and N be a natural number. Using pigeon-hole principle, show that there exists integers p and q such that 1 ≤ q ≤ N and

3. Let G = (V,E) be a graph where V is the vertex set and E is the edge set. A bijective mapping f : V → V is an automorphism if it has the property that (u,v) ∈ E ⇐⇒ (f(u),f(v)) ∈ E. Consider the following graph.
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Let A = {a1,a2,a3},B = {b1,b2,b3},M = {m1,m2,m3,m4}. Then, the vertex set of the above graph is V = A ∪ B ∪ M. Consider a bijective mapping g : A ∪ B → A ∪ B such that g(ai) ∈ {ai,bi} and g(bi) ∈ {ai,bi} for all i ∈ {1,2,3}, i.e., g maps the ordered pair [ai,bi] to either [ai,bi] (no swap) or [bi,ai] (swap).
Show that g can be extended to an automorphism f for the above graph if and only if the number of swaps performed by g is even.
4. An endomorphism of a ring R is a ring homomorphism φ : R 7→ R. Prove that φ : FP →7 Fp, φ(x) = xp is an endomorphism where p is a prime number.
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