System of Congruence problem

Find all solutions to the system of congruences: x ≡ 2 (mod 3) x ≡ 1 (mod 4) x ≡ 3 (mod 5)

Fermat's little theorem

Use Fermat's Little Theorem to compute 3^302 mod 5, 3^302 mod 7, and 3^302 mod 11. Use your results to find 3 ^ 302 mod 385 (note 385 = 5 . 7 . 11.)

Show that the binary expansion of a positive integer can be obtained from its hexadecimal expansion by translating each hexadecimal digit into a block of four binary digits.

Show that the binary expansion of a positive integer can be obtained from its hexadecimal expansion by translating each hexadecimal digit into a block of four binary digits.

Give a procedure for converting from the hexadecimal expansion of an integer to its octal expansion using binary notation as an intermediate step.

Give a procedure for converting from the hexadecimal expansion of an integer to its octal expansion using binary notation as an intermediate step.

Ordered Pais & Sets

1. a) Recall that ordered pairs must have the property that (x,y) = (u,v) if and only if x = u and y = v. Prove that {{x}, {x,y}} = {{u}, {u,v}} if and only if x = u and y = v. Therefore, although we know that (x,y) does not equal {x,y} , we can define the ordered pair (x,y) as the set {{x}, {x,y}}. b) Show by an ex ...continues

Sets

4. Let A = {a, {a}, {{a}}} B = {ø, {a}, {a, {a}}} C = {a} Be subsets of S = {ø, a, {a}, {{a}}, {a, {a}}}. Find a) A C b) B C’ c) A B d) ø B e) (B C) A f) A’ B g) {ø} B 5. Let A = {x | x is the name of a former president of the US} B = {Ada ...continues

Sets

8. Which of the following are true for all sets A, B, and C? a) B B = B b) (A’)’ = A c) (A – B) (B – A) = ø d) If A B = ø, then A is a proper subset of B e) B X A = A X B f) ø X A = ø g) ø {ø} = ø h) (A – B) (B – C) = A – C i) (A – C) (A – B) = A – (B C) 9. For any finite se ...continues

Prove the following statements using induction:

This problem set has nine statements that must be proven using induction. Please see the attachment, because summation signs, product signs, etc. can't be understood in text. Some of the questions in this problem set are: 3. Let A be an infinite set, then show that |P(A)| = 2^|A| 8. Let n > 1 be an integer. Then, show ...continues

Questions about ordered pairs, ordered triples, binary/unary operations, postfix notation

1. Recall that ordered pairs must have the property that (x,y) = (u,v) if and only if x = u and y = v. a) Prove that {{x}, {x,y}} = {{u}, {u,v}} if and only if x = u and y = v. Therefore, although we know that (x,y) does not equal {x,y} , we can define the ordered pair (x,y) as the set {{x}, {x,y}}. b) Show by an exa ...continues

Matrices

Find a matrix A such that ... See attached file for full problem description.