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Questions and Answers
What is the definition of mappings?
What is the definition of mappings?
- Mappings are the same as equivalences.
- Mappings can have multiple outputs for a single input.
- Mappings are always one-to-one relationships.
- A mapping associates each element of one set to a unique element of another set. (correct)
The two mappings $eta$ and $eta$ are equal if their domains are different.
The two mappings $eta$ and $eta$ are equal if their domains are different.
False (B)
What is a binary operation on a set?
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 $B$?
What is the symmetric difference of sets $A$ and $B$?
What conditions must hold for an algebraic structure to be considered a group?
What conditions must hold for an algebraic structure to be considered a group?
An algebraic structure is a semigroup if only associativity is satisfied.
An algebraic structure is a semigroup if only associativity is satisfied.
If $(G, \star)$ is a group and $\star$ is commutative, then $(G, \star)$ is an _____ group.
If $(G, \star)$ is a group and $\star$ is commutative, then $(G, \star)$ is an _____ group.
What is the general linear group of degree $n$?
What is the general linear group of degree $n$?
What does the Division Algorithm state for integers $a$ and $n$?
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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