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SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs BY: MISS FARAH ADIBAH ADNAN IMK

SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs BY: MISS FARAH ADIBAH ADNAN IMK. Chapter 1. 1.2 FUNCTIONS. CHAPTER OUTLINE: PART II. 1.2.1 DEFINITION OF FUNCTION. 1.2.2 SPECIAL TYPES OF FUNCTION. 1.2.3 INVERSE FUNCTION. 1.2.4 COMPOSITION OF FUNCTION. 1.2 Function.

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SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs BY: MISS FARAH ADIBAH ADNAN IMK

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  1. SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAsBY: MISS FARAH ADIBAH ADNANIMK Chapter 1

  2. 1.2 FUNCTIONS CHAPTER OUTLINE: PART II 1.2.1 DEFINITION OF FUNCTION 1.2.2 SPECIAL TYPES OF FUNCTION 1.2.3 INVERSE FUNCTION 1.2.4 COMPOSITION OF FUNCTION

  3. 1.2 Function 1.2.1 Definition of Function: • Let and be sets. A function from to , we write as , is an assignment of all elements in set to exactly one element of . • Symbols for the function, . • Sometimes write as • Set is called domain, and set is called range / image. • Image is often a subset of a larger set, called codomain. x y

  4. Example 1.1 Find the domain, range and codomain of .

  5. 1.2.2 Special Types of Functions: • ONE TO ONE / INJECTIVE • A function is said one to one, if and only if • Have a distinct images, at a distinct elements of their domain. • Eg:

  6. 2) ONTO / SURJECTIVE • Let a function from A to B, it is called onto if and only if for every element , there is an element . • Eg: refer textbook.

  7. 3) BIJECTION • Have both one to one and onto. • Eg: Let be the function from with Is is a bijection?

  8. 1.2.3 Inverse Functions: • Let be a function whose domain is the set , and the codomain is the set . Then the inverse function, has domain of the set Y and codomain of the set X, with the property: • The inverse function exists if and only if is a bijection.

  9. Example 1.2 1) Let be a function from {a,b,c} to {1,2,3} such that Is invertible? What is its inverse? 2) Let be the function from the set of integers such that . Is invertible? What is its inverse?

  10. 1.2.4 Composition of Functions: • Let be a function from the set A to the set B, and let be a function from the set B to the set C. The composition of the functions and , denoted by , is defined by: • The composition of cannot be defined unless the range of is a subset of the domain

  11. Example 1.3 Let be the function from the set {a,b,c} to itself such that Let be the function from the set {a,b,c} to the set {1,2,3} such that What is the composition of and , and what is the composition of and ?

  12. Example 1.4 Let and be the function from the set of integers defined by . What is the composition of and , and and ?

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