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Hilbert's epsilon-calculus used the letter ε ε to denote a value satisfying a predicate. If ϕ(x) ϕ (x) is any property, then εx. ϕ(x) ε x. ϕ (x) is a term t t such that ϕ(t) ϕ (t) is true, if such t t exists. One can define the usual existential and universal quantifiers ∃ ∃ and ∀ ∀ in terms of the ε ε quantifier:
y:= 7x + 2 y:= 7 x + 2. means that y y is defined to be 7x + 2 7 x + 2. This is different from, say, writing. 1 =sin2(θ) +cos2(θ) 1 = sin 2 (θ) + cos 2 (θ) where we are saying that the two sides are equal, but we are not defining "1" to be the expression " sin2(θ) +cos2(θ) sin 2 (θ) + cos 2 (θ) ". Basically, some people think that there ...
f: x ↦ y means that f is a function which takes in a value x and gives out y. f: N → N means that f is a function which takes a natural number as domain and results in a natural number as the result. Because you're wrong: the → and ↦ arrows mean different things. Also, W is not the set of positive numbers: that's R +.
For a closer similarity "≃" might mean a triangle almost congruent but only ROUGHLY similar, such as two triangles 3/4/5 and 3.1/4.1/5.1 while "≅" means congruent. Real life triangles use approximations and have rounding errors. 3/4 does not equal 3.1/4.1 but could be rough approximations for something already constructed.
Writing {\displaystyle x\in A} x\in A means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A". The expressions "A includes x" and "A contains x" are also used to mean set membership, however some authors use them to mean instead "x is a subset of A".
This notation means that you take the output of h h and use it as the input of f f. When we are working with a specific x x value, we can suggestively write f(h(x)) f (h (x)) instead. (f ∘ h)(x) = f(h(x)) = f(2 + 3x) = 1 2 + 3x. (f ∘ h) (x) = f (h (x)) = f (2 + 3 x) = 1 2 + 3 x. (Note: I only used z z as the variable for f f to avoid ...
20. "Much less than" is a qualitative assessment of comparative inequality. a ≪ b means that a is not only less than b, but muchly so by some convention. That convention is fairly arbitrary, or rather contextual; there is no fixed quantitative basis. Sometimes it is based on additive position, sometimes on multiplicative magnitude.
$\begingroup$ @Brian M. Scott What does it mean when each subscript is non-numerical, and is exactly the same? I have an equation that has a value T (temperature in Kelvin), subscripted gamma. It's used several times each the same way, no variation, and all other equations in the same family simply use T, unsubscripted. What might that mean ...
1. It also means (implicitly yet rigorously) that if Y Y is false then X X is false and if X X is false Y Y maybe false or true. In this sense ⇒ ⇒ is simply a logical connective like and, or, etc. A widespread abuse of notation is to tacitly assume that X X is true, which leads in very few cases to logical disasters.