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Questions and Answers
Define the meaning of the symbol $x\in S$, and provide an example of an expression utilizing this symbol.
Define the meaning of the symbol $x\in S$, and provide an example of an expression utilizing this symbol.
The symbol $x\in S$ denotes that $x$ belongs to the set $S$. For example, if $S = {1, 2, 3}$, then $2\in S$ is true, while $4\in S$ is false.
Explain the concept represented by the symbol $a\leq b$, and give an example of two numbers that satisfy this condition.
Explain the concept represented by the symbol $a\leq b$, and give an example of two numbers that satisfy this condition.
The symbol $a\leq b$ indicates that $a$ is less than or equal to $b$. For instance, $3\leq 5$ is true, while $7\leq 3$ is false.
What is the meaning of the symbol $\min{x, y}$, and provide an example of its application.
What is the meaning of the symbol $\min{x, y}$, and provide an example of its application.
The symbol $\min{x, y}$ represents the minimum value between $x$ and $y$. For example, $\min{3, 6} = 3.
Explain the concept denoted by $a_k + a_{k+1} + \cdots + a_m$, and provide a general expression for this concept.
Explain the concept denoted by $a_k + a_{k+1} + \cdots + a_m$, and provide a general expression for this concept.
Define the meaning of the symbol $n!$ and provide an example calculation.
Define the meaning of the symbol $n!$ and provide an example calculation.
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Study Notes
Mathematical Symbols and Meanings
- Z: Represents the set of integers.
- x ∈ S: Indicates that element x belongs to set S.
- x ∉ S: Indicates that element x does not belong to set S.
- Z+: Denotes the set of positive integers.
- N: Represents the set of whole numbers.
- a ≤ b: Signifies that a is less than or equal to b.
- a ≥ b: Signifies that a is greater than or equal to b.
- min{x, y}: Refers to the minimum value between x and y.
- max{x, y}: Refers to the maximum value between x and y.
- |x|: Represents the absolute value of x.
- ⌊x⌋: Denotes the floor of the real number x (greatest integer ≤ x).
- ⌈x⌉: Denotes the ceiling of the real number x (smallest integer ≥ x).
Summation and Factorials
- Σ (from i=m to n) ai: Represents the sum of the values of ai as index i runs from m to n.
- n!: Represents n factorial, the product of all positive integers up to n.
- (n choose k): Known as the binomial coefficient, represents the number of ways to choose k elements from a set of n elements.
Special Numbers
- Triangular number: Represents a value that can form an equilateral triangle.
- Square number: Represents a number that is the square of an integer.
- Pentagonal number: Represents a figurate number that can form a pentagon.
- Hexagonal number: Represents a figurate number that can form a hexagon.
- Tetrahedral number: Represents a figurate number that can fill a tetrahedron.
- Square pyramidal number: Represents a number that can form a pyramid with a square base.
- Pentagonal pyramidal number: Represents a number that can form a pyramid with a pentagonal base.
Other Symbols
- a div b: Refers to the integer division of a by b.
- a mod b: Refers to the remainder of the division of a by b.
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