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Given a set X, a relation R over X is a set of ordered pairs of elements from X, formally: R ⊆ { (x,y) | x, y ∈ X}. [2] [10] The statement (x,y) ∈ R reads "x is R-related to y" and is written in infix notation as xRy. [7] [8] The order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy.
The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional ...
In this example, the rule says: multiply 3 by 2, getting 6. The sets {A, B, C} and {X, Y} in this example are disjoint sets, but that is not necessary.The number of ways to choose a member of {A, B, C}, and then to do so again, in effect choosing an ordered pair each of whose components are in {A, B, C}, is 3 × 3 = 9.
The following proposition says that for any set , the power set of , ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.
If you grew up in a place where Christmas is celebrated, there’s a decent chance that you’ve heard the song — or at least the story — of Rudolph the Red-Nosed Reindeer. But it turns out ...
The shiny "red nose" that we now know so well was invented for the 1939 story Rudolph the Red Nosed Reindeer, written by Robert L May. His popularity would only continue to soar when the hit song ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
Key Points from 24/7 Wall St.: The maximum Social Security check in 2025 is $5,108. There’s a maximum check because there is a cap on how much income is used to calculate benefits.
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