Unary and binary set operations

A unary operation is an operation with only one operand while a binary operation is a calculation that combines two elements of operands.

1. Unary And Binary Set Operations By Prof. Liwayway Memije-Cruz

2. Set a well-defined collection of distinct objects. all the "things" that have common property. things grouped together with a certain property in common .

3. Set Symbols

4. Notation list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing : curly brackets { } are sometimes called "set brackets" or "braces ".

5. Numerical Sets Set of even numbers: {..., -4, -2, 0, 2, 4, ...} Set of odd numbers: {..., -3, -1, 1, 3, ...} Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...} Positive multiples of 3 that are less than 10: {3, 6, 9}

6. Why are Sets Important? Sets are the fundamental property of mathematics. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory and many more.

7. Universal Set It's a set that contains everything . Everything that is relevant to our question.

8. Equality Two sets are equal if they have precisely the same members. Example: Are A and B equal where: A is the set whose members are the first four positive whole numbers B = {4, 2, 1, 3} Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal! And the equals sign (=) is used to show equality, so we write : A = B

9. Subsets When we define a set, if we take pieces of that set, we can form what is called a subset. Example: the set {1, 2, 3, 4, 5 } A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc . But {1, 6} is not a subset, since it has an element (6) which is not in the parent set . A is a subset of B if and only if every element of A is in B .

10. Operand The operand is the object that is being worked on by an operation. Operations can be mathematical ones such as multiplication or addition, or they can be more sophisticated functions . In all computer languages, expressions consist of two types of components: operands and operators. Operands are the objects that are manipulated and operators are the symbols that represent specific actions. For example, in the expression 5 + x xand 5 are operands and + is an operator. All expressions have at least one operand.

11. Unary Set Operations In mathematics, a unary operation is an operation with only one operand, i.e. a single input. An example is the function f : A → A, where A is a set. The function f is a unary operation on A . Common notations are prefix notation (e.g. +, −, ¬), postfix notation (e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and superscripts (e.g. transpose AT). Other notations exist as well. For example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument .

12. Unary Operators In common arithmetic, the unary operators are negation, the reciprocal, and the absolute value. Negation involves reversing the sign of a number. For example, the negation of 4 is -4, and the negation of -23 is 23. The reciprocal involves dividing 1 by the number. Thus, the reciprocal of 4 is 1/4, and the reciprocal of -23 is -1/23. The absolute value involves reversing the sign of a number if it is negative, and leaving the number unchanged if it is 0 or positive. Thus, the absolute value of 4 is 4, and the absolute value of -23 is 23.

13. Unary negative and positive As unary operations have only one operand they are evaluated before other operations containing them. Here is an example using negation : 3 − − 2 Here, the first '−' represents the binary subtraction operation, while the second '−' represents the unary negation of the 2 (or '−2' could be taken to mean the integer −2). Therefore, the expression is equal to : 3 − (−2) = 5 Technically there is also a unary positive but it is not needed since we assume a value to be positive : (+2) = 2 Unary positive does not change the sign of a negative operation : (+(−2)) = (−2 ) In this case a unary negative is needed to change the sign : (−(−2)) = (+2)

14. Binary Operations

15. Binary Operation In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set. More formally, a binary operation is an operation of arity (the number of arguments or operands that the function takes) of two whose two domains and one codomain are the same set. Examples include the familiar elementary arithmetic operations of addition, subtraction, multiplication and division. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups.

16. Properties of Binary Operations

17. Commutativity of Binary Operations A binary operation is said to be commutative if a change in the order of the arguments results in equivalence . Example, multiplication on the real numbers is said to be commutative since ∀x,y ∈ R , x+y=y+x . However, there are examples where multiplication is not commutative. For example, if we are given two square matrices A and B, their product AB≠BA for all matrices A and B. In fact, AB=BA only for certain cases, hence it is important to note what sort of set we're talking about for binary operations.

18. Associativity of Binary Operations A binary operation is said to be associative if parentheses can be reordered and the result is equivalent . Example, addition is associative since ∀x,y,z∈R, (x+y)+z=x+(y+z), for example, (1 + 2) + 3 = 1 + (2 + 3).

19. Distributivity of Binary Operations Distributivity applies when we combined multiplication and addition . Example, on the real numbers ∀x,y,z∈R, z(x+y)=zx+zy. That is we have distributed the term z over the sum (x + y).

20. Identity Elements of Binary Operations An element e is said to be an identity element (or neutral element) of a binary operation if under the operation any element combined with e results in the same element . One common example can be seen in addition on real numbers when our identity element e is 0. That is x+e=x only when e = 0. The identity element is not always 0 though. In multiplication on real numbers, xe=x only when our identity element e is 1, and if A is an mxm square matrix then Ae=A only if e is the mxm identity matrix I mxm .

21. Inverses of Binary Operations For an element x, the inverse denoted x −1 when combined with x under the binary operation results in the identity element for that binary operation . Example , for addition on real numbers, the identity element is 0. Hence x+x −1 =0 only when our inverse is -x since ∀x∈R, x+(−x)=0 . For multiplication on real numbers, since our identity element is 1, then x⋅x −1 =1 only when our inverse is 1x. However, 0∈R, however 0 has no inverse, hence we say that x − 1 =1/x is a multiplicative inverse.

22. References http:// mathonline.wikidot.com/unary-and-binary-set-operations http://www.maths.manchester.ac.uk/~ mdc/MATH10101/2008-09/Ch8%20Groups%20Rings%20and%20Fields.pdf http:// mathonline.wikidot.com/associativity-and-commutativity-of-binary-operations https:// www.techopedia.com/definition/25643/unary-operator-c https:// whatis.techtarget.com/definition/unary https:// www.futurelearn.com/courses/maths-puzzles/0/steps/14011 https:// www.mathsisfun.com/sets/sets-introduction.html