Let X = {1,2,3} and let Y = {a,b}. [^)=Y f(gr) =f Let f: X→ Y be defined by f(1) = b, ƒ(2) = a, f(3) = b. (a) 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 of is the identity function on Y. Show that there is no function g: X → Y such that fog is the identity function on X. Suppose that X is a set with 1 element x. How many binary operations exist on X?
Let X = {1,2,3} and let Y = {a,b}. [^)=Y f(gr) =f Let f: X→ Y be defined by f(1) = b, ƒ(2) = a, f(3) = b. (a) 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 of is the identity function on Y. Show that there is no function g: X → Y such that fog is the identity function on X. Suppose that X is a set with 1 element x. How many binary operations exist on X?
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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Transcribed Image Text:a binary operation on S? Explain.
3.
= f(1)
a
fess=
b
Let X = {1,2,3} and let Y = {a,b}.
f(g) =f
(a)
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.
4.
Suppose that X is a set with 1 element x. How many binary operations exist on X?
We only have to define what x* x is ... and there is only one choice, namely, x. So there is
only one binary operation! And obviously, it is commutative!
(a)
Suppose X has two elements, î₁ and î2.
• Explain why there are 16 binary operations on X.
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.
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