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A Binary Operation on a Set S Is an Operation That Takes Two Elements of S As Input and Produces One Element of S As Output

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Definitions! • A binary operation on a set S is an operation that takes two elements of S as input and produces one element of S as output. • A set S is closed under an operation if whenever the inputs to the operation come from S, the output of the operation is in S too. ◦ [Note that in order for an operation to qualify as a binary operation on a set under the first definition above, the set must be closed under the operation, because the definition of binary operation requires that the output of the operation be in the set. If the set isn't even closed under the operation, then the operation does not qualify as a binary operation on the set, and none of the following definitions apply.] • A binary operation ? on a set S is commutative if a ? b = b ? a for all elements a and b in S. • A binary operation ? on a set S is associative if (a ? b) ? c = a ? (b ? c) for all elements a, b, and c in S. • Let S be a set with a binary operation ?. An element e in S is an identity element (or just an identity) if e ? a = a and a ? e = a for every element a in S. • Let S be a set with a binary operation ? and an identity element e. ◦ Let a be an element in S. If there exists an element b in S such that a ? b = e and b ? a = e, then the element a is invertible, and b is an inverse of a. ◦ If every element a in S is invertible, then the binary operation ? itself is called invertible. • A binary operation ? on a set S is idempotent if a ? a = a for every element a in S. • A set with a binary operation is called a magma. [Note that the set must be closed under the operation|otherwise the operation wouldn't qualify as a binary operation on the set!] • There are some special names for magmas that have additional properties. ◦ Semigroup: associativity. ◦ Monoid: associativity and identity. ◦ Group: associativity, identity, and invertibility. ◦ Abelian group: associativity, identity, invertibility, and commutativity. ◦ Semilattice: associativity, commutativity, and idempotence. ◦ Bounded semilattice: associativity, commutativity, idempotence, and identity. MAGMA associativity SEMIGROUP commutativity and identity idempotence MONOID SEMILATTICE invertibility identity GROUP BOUNDED SEMILATTICE commutativity ABELIAN GROUP 1. The set of positive integers under the operation of addition. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 2. The set of nonnegative integers under the operation of addition. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 3. The set of all integers under the operation of addition. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 4. The set of positive integers under the operation of subtraction. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 5. The set of all integers under the operation of subtraction. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 6. The set of integers under the operation of multiplication. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 7. The set of rational numbers under the operation of multiplication. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 8. The set of integers under the operation of division. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 9. The set of nonzero rational numbers under the operation of division. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 10. The set of positive integers under the operation of integer division: division where the remainder is thrown away. For example, under integer division, 38 ÷ 5 = 7, because the remainder of 3 is thrown away. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 11. The set of positive integers under the operation of exponentiation. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 12. The set of real numbers under the operation of exponentiation. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 13. The set of rational numbers whose denominators are 1 or 2, under the operation of addition. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 14. The set of rational numbers whose denominators are 1, 2, or 3, under the operation of addition. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice a+b 15. The set of real numbers in the interval [0; 1], under the operation h defined by a h b = 2 . closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice a+2b 16. The set of all real numbers under the operation i defined by a i b = 3 . closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice ab 17. The set of rational numbers under the operation defined by a b = a+b . closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 18. The set of positive rational numbers under the operation defined above. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 19. The set of rational numbers under the operation defined by a b = ab + 1. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice ab 20. The set of positive integers under the operation ~ defined by a ~ b = 2 . closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 21. The set of real numbers under the operation _ defined by a _ b = maxfa; bg. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 22. The set of real numbers under the operation ^ defined by a ^ b = minfa; bg. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 23. The set of positive integers under the operation N defined by a N b = gcd(a; b). closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 24. The set of positive integers under the operation H defined by a H b = lcm(a; b). closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 25. The set f0; 1; 2; 3; 4g under the operation of addition modulo 5, written ⊕5. Addition modulo 5 is done by adding the two numbers together and then taking the remainder when the sum is divided by 5. For example, 2 ⊕5 4 = 1, because 2 + 4 = 6, and the remainder when 6 is divided by 5 is 1. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 26. The set f0; 1; 2; 3; 4g under the operation of multiplication modulo 5, written ⊗5. Multiplication mod- ulo 5 is done by multiplying the two numbers together and then taking the remainder when the product is divided by 5. For example, 3 ⊗5 4 = 2, because 3 × 4 = 12, and the remainder when 12 is divided by 5 is 2. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 27. The set f1; 2; 3; 4g under the operation of multiplication modulo 5. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 28. The set f1; 2; 3; 4; 5g under the operation of multiplication modulo 6, written ⊗6. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 29. The set of integers under the operation / defined by a / b = a. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 30. The set of integers under the operation J defined by a J b = −a. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 31. The set of real numbers under the operation { defined by a { b = the least integer that is greater than a + b. closed commutative associative identity: invertible idempotent magma semigroup monoid group abelian group semilattice bounded semilattice 32. The set of states of the U.S., under the operation | defined by a | b = Kentucky. closed commutative associative identity:
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