Questions and Answers
What is the definition of mappings?
The two mappings $eta$ and $eta$ are equal if their domains are different.
False
What is a binary operation on a set?
A mapping from $A \times A$ to $A$.
What is the symmetric difference of sets $A$ and $B$?
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What conditions must hold for an algebraic structure to be considered a group?
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An algebraic structure is a semigroup if only associativity is satisfied.
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If $(G, \star)$ is a group and $\star$ is commutative, then $(G, \star)$ is an _____ group.
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What is the general linear group of degree $n$?
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What does the Division Algorithm state for integers $a$ and $n$?
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Study Notes
Mappings
- A mapping from set ( S ) to set ( T ) assigns each element of ( S ) to a unique element of ( T ).
- Set ( S ) is called the domain, while set ( T ) is the codomain.
- A mapping is synonymous with a function; if ( x \in S ), then ( \alpha(x) ) is the image of ( x ).
Equivalence of Mappings
- Two mappings ( \alpha ) and ( \beta ) are considered equal if their domains are equal and ( \alpha(x) = \beta(x) ) for every ( x ) in their domain.
- Symbolically represented as ( [\alpha = \beta] \implies [S_{\alpha} = S_{\beta} \wedge T_{\alpha} = T_{\beta} \wedge \alpha(x) \in T = \beta(x) \in T] ).
Binary Operation
- A binary operation on a set ( A ) is a mapping ( \alpha: A \times A \to A ).
Symmetric Difference
- Denoted by ( A \triangle B ), the symmetric difference of sets ( A ) and ( B ) is defined as ( (A \backslash B) \cup (B \backslash A) ).
- Equivalent to ( (A \cup B) \backslash (B \cap A) ); represents the union of two sets excluding their intersection.
Algebraic System / Mathematical System
- Generally refers to any set with operations defined on it that satisfy specific axioms or properties, though the definition was not provided in detail.
Groups / Semigroups / Monoid
- For a set ( G ) and a binary operation ( \star ):
- If ( \star ) is associative and there exists an identity element ( e ) in ( G ), ( (G, \star) ) is a group.
- If only associativity holds, ( (G, \star) ) is a semigroup.
- If associativity and identity element are present, ( (G, \star) ) is a monoid.
Abelian Groups
- If ( (G, \star) ) is a group and ( \star ) is also commutative, it is termed an abelian group.
General Linear Group of Degree ( n )
- The general linear group of degree ( n ), denoted ( GL(n, S) ), comprises ( n \times n ) invertible matrices under matrix multiplication.
- Inverses exist for matrices, and the identity element is the identity matrix.
The Division Algorithm
- For integers ( a ) and positive integer ( n ), there exist unique integers ( q ) and ( r ) such that ( a = qn + r ) with ( 0 \leq r < n ).
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Description
Explore key concepts in Abstract Algebra with these informative flashcards. Each card features important terms and their definitions, helping you grasp the essential principles of mappings and equivalence relations. Perfect for students looking to enhance their understanding of algebraic structures.
