please answer question 2, and 4 2. (al Determine all bijections from the HI, 2,

Question

please answer question 2, and 4

2. (al Determine all bijections from the HI, 2, 3I into ka, b, cl. b Determine all bijections from II, 2, 3I into la, b, c, dl. 3. Which of the following are one-toone, onto, orboth? R defined by fi(x) -x. a) f :R :Z Z defined by 2. N defined by f j, k) 2/3. (d) fa :P P defined by fa (n) n/21, where xisthe ceiling of t, the smallest integer greater than or equal to r. N defined by n (f) f :N Nx N defined by fo (n) 2n, 2n+ 4. Which of the following are injections sujections,orbijections on R,the setofreal numbers? X20

Explanation / Answer

2.

"In mathematics, a bijection (or bijective function or one-to-one correspondence) is a function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In formal mathematical terms, a bijective function f: X ? Y is a one to one and onto mapping of a set X to a set Y."

(a)

{1, 2, 3} -> {a, b, c}

The bijections are:

i. 1 -> a, 2 -> b, 3 -> c

ii. 1 -> a, 2 -> c, 3 -> b

iii. 1 -> b, 2 -> a, 3 -> c

iv. 1 -> b, 2 -> c, 3 -> a

v. 1 -> c, 2 -> a, 3 -> b

vi. 1 -> c, 2 -> b, 3 -> a

(b)

{1, 2, 3} -> {a, b, c, d}

As number of elements in both sets are different, no bijection is possible

4.

"In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain."

"In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if every element y in Y has a corresponding element x in X such that f(x) = y. "

(a)

Injection, Surjection, Bijection

(b)

not Injection, not Surjection, not Bijection

(c)

Injection, Surjection, Bijection

(d)

Injection, not Surjection, not Bijection

(e)

Injection, Surjection, Bijection

(f)

not Injection, Surjection, not Bijection

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