I need help with number 4 a = f()Let X = {1,2,3} and let Y = {a,b}.(a)f(g) =fLet f: X→Y be defined by f(1) = b, f(2)= a, f(3) = b.Show that there is a function g: Y→ X such that fog is the identity function on Y.Show that there is no function g: Y→ X such that go f is the identity function on X.Let f: Y→ X be defined by f(a) = 3 and f(b) = 2.(b)Show that there is a function g: X→ Y such that go f is the identity function on Y.Show that there is no function g: X → Y such that fog is the identity function on X.a binary operation on S? Explain.3.4.Suppose that X is a set with 1 element x. How many binary operations exist on X?and there is only one choice, namely, x. So there isWe only have to define what x * x isonly one binary operation! And obviously, it is commutative!Suppose X has two elements, ₁ and 2.on X.(a)• Explain why there are 16 binary ope. Explain why exactly 8 of them are commutative.(b)Suppose X has n elements.• How many binary operations are there on X? Explain.How many of these binary operations are commutative? Eplain.

Answered: with 1 element x. How *x is ... and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

I need help with number 4

a = f()
Let X = {1,2,3} and let Y = {a,b}.
(a)
f(g) =f
Let f: X→Y be defined by f(1) = b, f(2)= a, f(3) = b.
Show that there is a function g: Y→ X such that fog is the identity function on Y.
Show that there is no function g: Y→ X such that go f is the identity function on X.
Let f: Y→ X be defined by f(a) = 3 and f(b) = 2.
(b)
Show that there is a function g: X→ Y such that go f is the identity function on Y.
Show that there is no function g: X → Y such that fog is the identity function on X.
a binary operation on S? Explain.
3.
4.
Suppose that X is a set with 1 element x. How many binary operations exist on X?
and there is only one choice, namely, x. So there is
We only have to define what x * x is
only one binary operation! And obviously, it is commutative!
Suppose X has two elements, ₁ and 2.
on X.
(a)
• Explain why there are 16 binary ope
. Explain why exactly 8 of them are commutative.
(b)
Suppose X has n elements.
• How many binary operations are there on X? Explain.
How many of these binary operations are commutative? Eplain.

Expert Solution

Step 1

Here, in the question it is been asked about the binary operations on $X$ . Binary operations are sometimes commutative. Binary operation mean two operations done on a variable. We have to find the binary opeartions occuring on the system.

In mathematics, a boolean operation is commutative if changing the order of the operands doesn't change the result. it's a fundamental property of the many binary operations, and lots of mathematical proofs rely on it. On the set of real numbers R, subtraction, that is, f(a, b) = a − b, could be a binary arithmetic operation which isn't commutative since, in general, a − b ≠ b − a. it's also not associative, since, in general, a − (b − c) ≠ (a − b) − c; as an example, 1 − (2 − 3) = 2 but (1 − 2) − 3 = −4.

For any number of elements the set of binary operations are: $n^{n^{2}}$