Identifying universal, equal and equivalent sets,

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1. Identifying universal sets , equal and equivalent sets , joint and disjoint sets Ms. G. Martin

2. Universal Sets a universal set is a set which contains all objects or all elements under consideration , including itself and denoted by capital U.   Example 1. A = {x / x is a vowel in the English alphabet} B = {x / x is a consonant in the English alphabet} Given set A and B, universal set is the set of all letter in the English alphabet. Thus, U = {x / x is a letter in the English alphabet}

3. Example 2. Let C = {1, 3, 5, 7, …} D = {2, 4, 6, 8, …} Given the sets C and D, the universal set is the set of counting numbers. Thus, U = {1, 2, 3, 4, 5, 6, 7, …} or U = {x / x is a counting number}

4. Equal and Equivalent Sets Two sets are equal if they have the same elements. We write A = B. Two sets are equivalent if they have the same cardinality. That is n(A) = n(B). We write A B .  

5. Equal Sets Example 1. A = {x / x is s first 5 multiples of 5} B = {2, 4, 6, 8, 10} A and B are equal sets because they have exactly the same elements. Thus , A = B Example 2. C = { x / x is a whole number less than 9} D = {1, 2, 3, 4, 5, 6, 7, 8} Thus , C = D

6. Equivalent Sets Example 1. C = {1, 2, 3, 5, 7} D = {a, e, i , o, u} C and D are equivalent because n(C) = 5 and n(D) = 5. C and D have the same cardinality. Thus, C D Example 2. E = {apple, mango, banana} F = {x / x is a multiple of 5 less than 17} Thus, E F  

7. Joint and Disjoint Sets Sets having common elements are called joint sets . Sets with no common elements are called disjoint sets . Examples: M = {1, 3, 5, 7, 9} and N {2, 4, 6, 8, 10} are disjoint sets 2. P = { 2 , 5 , 6, 9, 13} and P = { 2 , 3, 5 , 8, 12} are joint sets because they have common elements 2 and 5.