List all the bijections from a into b.
a) a={q, r,s} and b={2,3,4}
b) a={1,2,3,4} and b={5,6,7,8}

The list of bijections from A into B are shown below.

Step-by-step explanation:

A function f is called one-to-one or injective, if and only if

for all x and y in the domain of f.

A function f from X to Y is called onto or surjective, if and only if

for every element y∈Y there is an element x∈X with f(x)=y.

If a function is one-one and onto, then it is called bijective.

Part (a):

A={q,r,s} and B={2,3,4}

We need to find all the bijections from A into B.

(1)

(2)

(3)

(4)

(5)

(6)

Part (b):

A={1,2,3,4} and B={5,6,7,8}

We need to find all the bijections from A into B.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240