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SETS-AND-SUBSETS.pptx

• 2. Four Basic Concepts SETS AND ITS BASIC OPERATIONS RELATIONS FUNCTIONS BINARY OPERATIONS
• 3. Sets and its basic operations Four Basic Concepts LESSON 1
• 4. DEFINITION In Math, sets are a collection of well- defined objects or elements. A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set in a curly bracket {…}.
• 5. EXAMPLES 1. A set of counting numbers from 1 to 10. 2. A set of English alphabet from a to c. 3. A set of integers. 1. A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 2. B = {a, b, c} 3. C = {.... -3, -2, -1, 0, 1, 2, 3, ...}
• 6. x S x S "x is an element of S" "x is not an element of S" What is an element of a set?
• 7. The language of Sets The notation means that an item belongs to a set. 1 A 3 A 6 A Pertains to each member of the set. Illustration Say A = {1, 2, 3, 4, 5} 5 A ELEMENT OF A SET
• 8. Terminologies of Sets Unit set is a set that contains only one element. UNIT SET Illustration A = {1} B= {c} C= {banana}
• 9. Terminologies of Sets Empty or null set is a set that has no element. EMPTY SET or NULL SET; Illustration A = {} A set of seven yellow carabaos.
• 10. Terminologies of Sets Finite set is a set which elements are countable. FINITE SET Illustration A = {1, 2, 3, 4, 5} B = {a, b, c, d}
• 11. Terminologies of Sets Infinite set is a set which elements are not countable or has no end. INFINITE SET Illustration Set of counting numbers A = {.... -3, -2, -1, 0, 1, 2, 3, ...}
• 12. Terminologies of Sets Total number of the elements in a set. CARDINAL NUMBER; n Illustration A = {2, 4, 6, 8} B = {a, c, e} n = 4 n = 3
• 13. Two sets are said to be equal iff the have equal number of cardinality and the elements are identical. EQUAL SET Illustration A = {1, 2, 3, 4, 5} B = {3, 5, 2, 1, 4} A and B are equal sets Terminologies of Sets
• 14. Two sets are said to be equivalent if the Sets have the exact number of elements. There is a 1-1 correspondence. EQUIVALENT SET Illustration A = {1, 2, 3, 4, 5} B = {a, b, c, d, e} A and B are equivalent sets Terminologies of Sets
• 15. Universal set is the set of all elements under discussion UNIVERSAL SET; U Illustration A set of an English alphabet U = {a, b, c, d, ...z} Terminologies of Sets
• 16. Two sets, say A and B, are said to be joint iff they have common element/s. JOINT SETS Illustration A = {1, 2, 3} B = {2, 4, 6} A and B are joint sets since they have a common element, 2. Terminologies of Sets
• 17. Two sets, say A and B, are said to be disjoint iff they are mutually exclusive or they don't have common element/s. DISJOINT SETS Illustration A = {1, 2, 3} B = {4, 6, 8} A and B are disjoint sets since they don't have a common element. Terminologies of Sets
• 18. TWO WAYS OF DESCRIBING A SET It is done by listing or tabulating the elements of the set. ROSTER or TABULAR METHOD Illustration A = {2, 4, 6} B = {a, b, d, e, h, i} C = {blue, red, yellow}
• 19. It is done by stating or describing the common characteristics of the elements of the set. We use the notation A = {x l x ...} RULE or SET-BUILDER METHOD Illustration A = {x l x is a counting number from 1 to 5} B = {x l x English alphabet} TWO WAYS OF DESCRIBING A SET
• 21. A subset, A B, means that every element of A is also an element of B. If x A then, x B SUBSETS In particular, every set is a subset of itself. A A
• 22. SUBSETS A subset is called a proper subset, A is a proper subset of B, if there is at least one element of B that is not in A. A B A = {1, 3, 5, 7} Illustration B = {1, 3, 5, 7, 9, 11} every element of A is an element of B there is at least 1 element in B that is not an element of A
• 23. The empty or null set, {} or , is a subset of every set. How many subsets are there in the set Illustration A = {1, 2, 3} The number of subsets of a given set is given by , n is the number of elements of the given set. 2 n List down all the subsets of A. SUBSETS
• 24. Solution: {1} 2 n List of subsets: Number of elements = 3 = 2 3 = 8 A has 8 subsets. With one element {2} {3} With two elements {1, 2} {1, 3} {2, 3} With three elements {1, 2, 3} With no element {} or SUBSETS
• 25. ORDERED PAIR "b" Given elements a and b, the symbol (a, b) denotes the ordered pair consisting of a and b together. "a" first element second element two ordered pairs (a, b) and (c, d) are equal iff a = c and b = d
• 26. ORDERED PAIR (a, b) = (c, d) Symbolically, means that a = c and b = d Illustration If (a, b) = (3, 2) then a = 3 and b = 2 Find x and y if (4x + 3, y) = (3x + 5, -2) 4x + 3 = 3x + 5 y = -2 4x - 3x = 5 - 3 x = 2
• 27. OPERATION ON SETS A B UNION OF SETS The union of sets A and B, denoted by is defined as A B = {x l x A or x B} Illustration If A = {1, 2, 3} and B = {4, 5}, then A B = {1, 2, 3, 4, 5} If A = {1, 2, 3} and B = {1, 2, 4, 5}, then A B = {1, 2, 3, 4, 5}
• 28. A B UNION OF SETS Illustration OPERATION ON SETS
• 29. The intersection of sets A and B, denoted by A B = {x l x A and x B} A B = {1, 2} A B = {} or A B INTERSECTION OF SETS is defined as Illustration If A = {1, 2, 3} and B = {4, 5}, then If A = {1, 2, 3} and B = {1, 2, 4, 5}, then OPERATION ON SETS
• 30. A B INTERSECTION OF SETS Illustration OPERATION ON SETS
• 31. The difference of sets A and B, denoted by A - B = {x l x A and x B} A - B = {3} A - B = {1, 2, 3} A - B DIFFERENCE OF SETS is defined as Illustration If A = {1, 2, 3} and B = {4, 5}, then If A = {1, 2, 3} and B = {1, 2, 4, 5}, then OPERATION ON SETS
• 32. A - B DIFFERENCE OF SETS Illustration OPERATION ON SETS
• 33. For set A, the difference U - A, where U is the universal set is called the complement of A, denoted by . Thus it is the set of everything that is not in A. A COMPLEMENT OF SETS Illustration Let U = {a, e, i, o u} and A = {a, e}, then c or A' Ac = {i, o, u} OPERATION ON SETS
• 35. Given sets A and B, the cartesian product of A and B, denoted by and read as "A cross B", is the set of all ordered pair (a, b), where a is in A and b is in B. A x B CARTESIAN PRODUCT = {(a,b) l a A and b B} A x B Note that A x B is not equal to B x A OPERATION ON SETS
• 36. CARTESIAN PRODUCT, cont'd A x B = {(1, a), (1, b), (2, a), (2, b)} Illustration If A = {1, 2, } and B = {a, b}, what is A x B? what is B x A? B x A = {(a, 1), (a, 2), (b, 1), (b, 2)} OPERATION ON SETS
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