Proof problem

Need help in determining the following proof. (See attached file for full problem description) --- Thm 11.1.2 (the pigeonhole principle): Suppose that f:X Y is a function between non-empty finite sets such that |X| > |Y|. Then f is not an injection, i.e. there exist distinct elements x1 and x2 E (epsilon) X such that f(x1) = f(x2). ---

Advanced math proof exercise

Need help in determining the following proof exercises. Please use general logic notation such as U. (See attached file for full problem description) --- Proposition 10.2.1: (the addition principle) Suppose that X and Y are disjoint finite sets. Then X U Y is finite and |X UY| = |X| + |Y|. Corollary 10.2.2: For a positive integer n, suppose that X1, X2….,Xn is a collection of n pairwise disjoint finite sets (i.e. i does not = j => Xi Xj = empty set) Then X1 U X2 U….U Xn = U ( lim from n to i=1) Xi is a finite set and |X1 U X2 U……U Xn| = |X1| + |X2|+….|Xn|. ---

Number Theory. 400 level. Introductory Course in Undergraduate.

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)

Number Theory. 400 level. Introductory Course in Undergraduate.

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)

Number Theory. 400 level. Introductory Course in Undergraduate.

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)