Mathematics Preliminary Notions Chapter 1

Questions and Answers

What is the result of applying the mapping o to the element Xx1?

X30

X45 (correct)

X35

What type of mapping is represented by the composition ooT?

It is an identity mapping of S into itself. (correct)

It is a mapping from T back into S.

It is a one-to-one mapping of S into T.

It is not defined.

If t acts on an element from set T, what will the result (m,n)t yield when m is even?

0 (correct)

m - n

m + n

Lif m is odd

What is the consequence of applying the mapping t followed by o to an integer n in the defined context?

It gives a new integer by adding n - 1 to the sum. Signup and view all the answers

What operation cannot be defined in the mapping o°t?

From T to T Signup and view all the answers

Which of the following correctly describes the mapping t defined from T to U?

It returns Lif if n is even and 0 if n is odd. Signup and view all the answers

When applying the mapping t°o, which of the following results can be inferred?

It cannot be computed as they are mapped differently. Signup and view all the answers

What is the outcome of applying the mapping o to the real number s = 2 + 4?

It yields 2. Signup and view all the answers

What can be inferred about the relationship between the elements w and W-¢ from the discussion?

They represent distinct elements in G. Signup and view all the answers

Which of the following statements about the cyclic group of order n is true?

It includes elements a' where a^n = e. Signup and view all the answers

What does the set G consist of, as defined in the context of finite mappings of S?

Mappings that only move a finite number of elements of S. Signup and view all the answers

According to the definitions provided, what is the order of the group G referenced?

It has six distinct elements. Signup and view all the answers

In the context of the group operation defined, what can be said about the product of any two elements o and t in G?

The product o:t is closure under the operation. Signup and view all the answers

Which of the following statements describes W given its role in relation to d: and y~'?

W plays a critical role in producing distinct outputs from y~'. Signup and view all the answers

What is the geometric realization of the cyclic group of order n described?

A rotation about a center point. Signup and view all the answers

What distinguishes the groups defined in terms of mappings from those that do not?

Mappings that only affect a limited, finite number of elements. Signup and view all the answers

What does the mapping defined as t:S x T > S where (a, b)t = a represent?

The projection of the Cartesian product S x T onto the set S. Signup and view all the answers

What can be inferred about the projection t defined from S x T to T?

It selects the second element of each pair in S x T. Signup and view all the answers

If S = {x1, x2}, how many elements would S*, the set of subsets of S, contain?

4 Signup and view all the answers

What does the term 'image of S under t' refer to?

The range of the mapping t from S to T. Signup and view all the answers

What is the condition for the mapping t:S → T to be considered 'onto'?

Each element in T must be the image of at least one element in S. Signup and view all the answers

In Example 1.2.8, what value is assigned to n when n is odd?

O Signup and view all the answers

Which of the following statements about the inverse image of t is false?

Each element in T must have an inverse image. Signup and view all the answers

In the context of equivalence relations, how is the mapping t:S → T defined?

It assigns each element its quotient class. Signup and view all the answers

What property does the element represented by the matrix possess in the group G?

It acts as an identity element. Signup and view all the answers

If 'a', 'b', 'c', and 'd' are constants in the group, what condition is used to verify the membership of a matrix in G?

$ad - bc eq 0$ Signup and view all the answers

Which of the following is true about the inverse of the matrix within G?

It can be represented by a specific matrix form. Signup and view all the answers

In the context of the group G, what does the expression $ad - bc eq 0$ imply?

The matrix has a non-zero determinant. Signup and view all the answers

What is the importance of verifying that $ad - bc eq 0$ for the elements of G?

It confirms that all elements produce a unique inverse. Signup and view all the answers

Which of the following statements about G is true?

It includes inverses for each element. Signup and view all the answers

In general group theory, what must hold true for any element within a group to qualify as a group?

There must exist an identity element and inverses for all elements. Signup and view all the answers

What mathematical operation is primarily discussed in relation to the group G?

Multiplication of matrices. Signup and view all the answers

What are trivial subgroups characterized by?

They do not exhibit interesting properties within a group. Signup and view all the answers

In the example where G is the group of integers under addition and H consists of multiples of 5, what type of subgroup is H?

Nontrivial subgroup Signup and view all the answers

What is true about cyclic groups in relation to abelian groups?

Cyclic groups are a special case of abelian groups. Signup and view all the answers

What does (W) represent in the context of a group G and subset W?

The set of all elements in G that are products of elements in W. Signup and view all the answers

When is a group G said to be cyclic?

When G can be generated by a single element. Signup and view all the answers

Which of the following subsets is a subgroup of the group of nonzero real numbers under multiplication?

Set of positive rational numbers. Signup and view all the answers

What can be concluded about H(x%) and H(x₁) when x ≠ y for a set S under one-to-one mappings?

Their intersection may result in an empty set. Signup and view all the answers

What is the nature of the subgroup generated by a single element a in a group G?

It consists of all powers of a. Signup and view all the answers

Study Notes

Set Mappings and Projections

Defined mapping ( t: S \times T \to S ) is the projection onto ( S ).
For a projection ( t(a, b) = a ), it extracts the first element from the Cartesian product ( S \times T ).
Analogously, projection onto ( T ) can be defined.

Subsets and Power Sets

( S^* ) represents the power set of ( S ), consisting of all subsets of ( S ).
For example, if ( S = {x_1, x_2} ), then ( S^* = {\emptyset, S, {x_1}, {x_2}} ).
The relation between ( S ) and ( S^* ) reveals interesting properties.

Equivalence Relations and Classes

If ( S ) has an equivalence relation, ( T ) can be defined as the set of equivalence classes.
Mapping ( t: S \to T ) sends each element ( s ) to its equivalence class ( cl(s) ).

Inverse Image and Onto Definitions

The inverse image of an element ( t ) with respect to ( T ) is the set of all elements in ( S ) that map to ( t ).
Mapping ( t ) is onto ( T ) if every ( t \in T ) has a pre-image in ( S ) such that ( t = st ).

Example Functions and Mappings

Various example mappings ( o: S \to T ) demonstrate functions like ( m \mapsto (m - 1, 1) ) changing elements of ( S ) into tuples.
Identity mapping ( o \circ o ) can sometimes yield a function returning the original argument.

Cyclic Groups and Their Properties

A cyclic group ( G ) of order ( n ) consists of elements ( a^i ) where ( i = 0, 1, ..., n - 1 ) with ( a^n = e ) being the identity.
Geometric interpretation involves rotations on a circle.

Subgroups and Their Construction

Subgroup ( H ) of integers under addition includes multiples of a number ( n ), demonstrating a structured collection of elements.
The cyclic subgroup generated by an element ( a ) in group ( G ) is denoted by ( (a) ), containing all integer powers of ( a ).

Group Properties and Operations

Operation ( G ) under the composite of mappings preserves group properties, ensuring closure and identity existence.
Example matrices show group elements defined through finite movements maintain group structure.

Real Number Groups

The group of non-zero real numbers under multiplication forms a basis for rational numbers as a subgroup.
Real numbers under addition allow integers to form a subgroup, illustrating subgroup relationships in different contexts.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Explore the preliminary notions in mathematics through this quiz. Delve into concepts such as product definitions, projections, and examples involving sets. Test your understanding of the fundamental ideas that underpin set theory and mathematical relations.