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 sn=-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.