Reference no: EM132625360

Question 1.

a) Prove that the following statement is true:

If a is any even integer and b is any oddinteger then a² + 2ab + b² is odd.

b) Find a counter example to disprove the following:

If ris a negative integer and s is not a negative integer, then the product rsis negative.

c) Given that √8 is irrational, prove that √8+ 0.5 is irrational.

Question 2.

a) Show that M: Z→Z defined by M(x)=(2x+5) mod 7 is a function and determine if it is one-to-one or onto.

b) Use the method of mathematical induction to prove that for each integer n≥1,
5+9+13+?+(4n+1)=2n²+3n.

Question 3.

A sequence is defined recursively by s_n=-s_(n-1)+3, for all integers n≥1,s_0=1. Write down the first four elements and develop an explicit formula for the sequence.

Question 4.

a) Prove the following statement using set identities:

For all sets A,B and C,(A-C)∩(B-C)∩(A-B) = Φ.

(Remember to cite which set identity you are using at each step.)

b) Prove that the set of all integers Z has the same cardinality as P, the set of all integers which are prime.

c)  Let A be the set of all lines in the plane. A relation Ω is defined on Aby

For all l₁, l₂∈A, l₁ Ω l₂ <=> l₁intersectsl₂.

Determine whether Ω is reflexive, symmetric, transitive, or none of these.