BRIO Builder Construction Set - STEM Learning, Building And Educational Toys Ages 3 Year Olds Up

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The following examples illustrate particular sets defined by set-builder notation via predicates. In each case, the domain is specified on the left side of the vertical bar, while the rule is specified on the right side. In each preceding example, each set is described by enumerating its elements. Not all sets can be described in this way, or if they can, their enumeration may be too long or too complicated to be useful. Therefore, many sets are defined by a property that characterizes their elements. This characterization may be done informally using general prose, as in the following example. A domain E can appear on the left of the vertical bar: [5] { x ∈ E ∣ Φ ( x ) } , {\displaystyle \{x\in E\mid \Phi (x)\},} x ∣ x ∈ E and Φ ( x ) } or { x ∣ x ∈ E ∧ Φ ( x ) } . {\displaystyle \{x\mid x\in E{\text{ and }}\Phi (x)\}\quad {\text{or}}\quad \{x\mid x\in E\land \Phi (x)\}.} In general, it is not a good idea to consider sets without defining a domain of discourse, as this would represent the subset of all possible things that may exist for which the predicate is true. This can easily lead to contradictions and paradoxes. For example, Russell's paradox shows that the expression { x | x ∉ x } , {\displaystyle \{x~|~x\not \in x\},} although seemingly well formed as a set builder expression, cannot define a set without producing a contradiction. [6]

The ∈ symbol here denotes set membership, while the ∧ {\displaystyle \land } symbol denotes the logical "and" operator, known as logical conjunction. This notation represents the set of all values of x that belong to some given set E for which the predicate is true (see " Set existence axiom" below). If Φ ( x ) {\displaystyle \Phi (x)} is a conjunction Φ 1 ( x ) ∧ Φ 2 ( x ) {\displaystyle \Phi _{1}(x)\land \Phi _{2}(x)} , then { x ∈ E ∣ Φ ( x ) } {\displaystyle \{x\in E\mid \Phi (x)\}} is sometimes written { x ∈ E ∣ Φ 1 ( x ) , Φ 2 ( x ) } {\displaystyle \{x\in E\mid \Phi _{1}(x),\Phi _{2}(x)\}} , using a comma instead of the symbol ∧ {\displaystyle \land } .When it is desired to denote a set that contains elements from a regular sequence, an ellipsis notation may be employed, as shown in the next examples:

The vertical bar (or colon) is a separator that can be read as " such that", "for which", or "with the property that". The formula Φ( x) is said to be the rule or the predicate. All values of x for which the predicate holds (is true) belong to the set being defined. All values of x for which the predicate does not hold do not belong to the set. Thus { x ∣ Φ ( x ) } {\displaystyle \{x\mid \Phi (x)\}} is the set of all values of x that satisfy the formula Φ. [4] It may be the empty set, if no value of x satisfies the formula.x ∈ R ∣ x > 0 } {\displaystyle \{x\in \mathbb {R} \mid x>0\}} is the set of all strictly positive real numbers, which can be written in interval notation as ( 0 , ∞ ) {\displaystyle (0,\infty )} .

In cases where the set E is clear from context, it may be not explicitly specified. It is common in the literature for an author to state the domain ahead of time, and then not specify it in the set-builder notation. For example, an author may say something such as, "Unless otherwise stated, variables are to be taken to be natural numbers," though in less formal contexts where the domain can be assumed, a written mention is often unnecessary. displaystyle \{7,3,15,31\}} is the set containing the four numbers 3, 7, 15, and 31, and nothing else.x ∈ R ∣ | x | = 1 } {\displaystyle \{x\in \mathbb {R} \mid |x|=1\}} is the set { − 1 , 1 } {\displaystyle \{-1,1\}} . This set can also be defined as { x ∈ R ∣ x 2 = 1 } {\displaystyle \{x\in \mathbb {R} \mid x