Question #185737

Consider the following two sets A & B: 

A= {4, 8, 12, 16, … }

B = {1, 3, 5, 7, 9, … }

Let  be a function from z × z to , such that f(m,n) = (m*m)-(n*n).

i) Show that every element of the set A has a preimage under the function f. 

Type/Insert your answer here! 

ii) Show that every element of the set B has a preimage under the function f. 

Type/Insert your answer here!


1
Expert's answer
2021-05-07T14:32:59-0400

Given:

Consider the following two sets A & B:

A= {4, 8, 12, 16, … }

B ={1, 3, 5, 7, 9, … }

Let f be a function from z × z to , such that f(m,n)=(mm)(nn)f(m,n) = (m*m)-(n*n) .

1) Show that every element of the set A has a preimage under the function f.

Solution:

aA=>a=4l,lNf(l+1,l1)=(l+1)2(l1)2=4l=a\forall a\in A=>a=4*l,l\in N\\ f(l+1,l-1)=(l+1)^2-(l-1)^2=4*l=a

2)Show that every element of the set B has a preimage under the function f. 

Solution:

bB=>b=2l+1,lNor0f(l+1,l)=(l+1)2l2=2l+1=b\forall b\in B=>b=2*l+1,l\in N or 0\\ f(l+1,l)=(l+1)^2-l^2=2*l+1=b

Answer:\\

(1) for aA preimage(a4+1,a41)\ a\in A \ preimage (\frac{a}{4}+1,\frac{a}{4}-1)

(2) for bB preimage(b12+1,b12)\ b\in B \ preimage (\frac{b-1}{2}+1,\frac{b-1}{2})


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