If N and K are submodules of M. (i) The kernel of the natural homomorphism N--> M/K is (N+K)/K. (ii) The kernel of the natural homomorphism N --> M/K is N/(N intersection K). (iii) If N is contained in K then the kernel of the natural homomorphism N/N --> M/K is K/N. O Both (i) and (ii) are true and (iii) is not true. O Both (ii) and (iii) are true and (i) is not true. O Both (i) and (iii) are true and (ii) is not true. O All (i),(ii) and (iii) are true.

Please help with both parts b and c: (will leave a like)

Determine the groups of quadratic residues Qn for all n in the range 10 ≤ n ≤ 20.

1. (a) (b) (c) (d) Prove or disprove that, for any universal set U and predicates P and Q, [3x = U, P(x) ^ Q(x)] → [3r EU, P(x)) ^ (3x = U, Q(x))] Prove or disprove that, for any universal set U and predicates P and Q, [3r EU, P(x)) ^ (3xU, Q(x))] → [r U, P(x) ^ Q(x)] Prove or disprove that, for any universal set U and predicate P [3r € U, P(x)] → [Vr € U, P(x)] Prove or disprove that, for any universal set U and predicate P [VxU, P(x)] → [3r € U, P(x)]

Q1// Let Ai be even permutation in Sn. Then * O(An)=n!/2 إضافة ملف Q2// Let Hi family of subgroups of (G, *). Prove that the intersection of Hi is also *.subgroup إضافة ملف