4.8. Set Operations. There are five main set theoretic operations, one corresponding to each of the logical connectives. Name

Transcription

1 4.8. Set Operatins There are five main set theretic peratins, ne crrespnding t each f the lgical cnnectives. Set Operatin Name Lgical nnective Name mplement ~ P Negatin «Unin P Q Disjunctin» Intersectin P Q njunctin Õ Subset P Q nditinal Equality P Q P Q icnditinal Equivalence The set peratins can be defined in terms f the crrespnding lgical peratins. This means that each f the tautlgies prved by truth tables fr the lgical cnnectives will have a crrespnding therem in set thery. WUT121 Lgic 148

2 We have seen hw the lgical cnditinal peratr, P Q is related t subset, bicnditinal peratr, and hw the lgical P Q r equivalence, P Q is related t set equality,. The fllwing sectins will cver the three remaining set peratins: cmplement, unin and intersectin. In ur discussin f set thery, we will let U be a fixed set and all ther sets, whether dented,,, etc, will be subsets f U. In ther wrds,,, U. Thus, each result shuld start with a statement similar t Let,, be subsets f a universal set U r Let,, U Definitin: mpliment Let U be a universal set, and let U. Then the cmplement f, dented by, is given by ~ x } x } : :. Ntes. 1. U \, and c are als used fr in sme bks. 2. If the set U is fixed in a discussin, then is smetimes written as { x : x } WUT121 Lgic 149

3 Example: The shaded area in the fllwing Venn diagram depicts : Exercises: Let U. Write dwn fr the fllwing sets: { 1, 2, 3} { x : x is even} { x : x > 0 x < 0} WUT121 Lgic 150

4 Definitin: Unin Let and be subsets f a universe U. Then the unin f and, dented by, is given by x x } :. Example: The shaded area in the fllwing Venn diagram depicts : Exercises: Let U. Write dwn fr the fllwing sets: {} 1 and { 2}. is the set f all even integers, is the set f all dd integers. WUT121 Lgic 151

5 { x : 0 x 2} and { x : 1 x 3} If U and U, is it true that U? Definitin: Intersectin Let and be subsets f a universe U. Then the intersectin f and, dented by x x } :., is given by Example: The shaded area in the fllwing Venn diagram depicts : WUT121 Lgic 152

6 Exercises: Let U. Write dwn fr the fllwing sets: { 1, 2, 3, 5} and { 1, 4, 5, 6}. is the set f all even integers, is the set f all dd integers. { x : 0 x 2} and { x : 1 x 3} If U and U, is it true that U? Definitin: Difference Let and be subsets f a universe U. Then the difference f and, dented by x x } :., is given by WUT121 Lgic 153

7 Example: The shaded area in the fllwing Venn diagram depicts : Ntes. 1. The difference f is smetimes called the relative cmplement f in. 2. If we let U U, then we have : x U x } : x } 3. Using Definitins fr cmplement and intersectin, we can simplify the definitin f difference as fllws: : x x } : x x } WUT121 Lgic 154

8 Exercises: Let U. Write dwn fr the fllwing sets: { 1, 2, 3, 5} and { 1, 4, 5, 6}. integers.. is the set f all even integers, is the set f all dd { x : 0 x 2} and { x : 1 x 3} If U and U, is it true that U? Let U, { 1, 2, 3}, { 2}, { 2, 3, 4} [ 0, 1] { x : 0 1} D x.write dwn: and D D D WUT121 Lgic 155

9 Definitin: Disjint sets Let and be subsets f a universe U. Then and are said t be disjint if. Example: The fllwing Venn diagram depicts disjint sets and : Nte. Disjint sets have n elements in cmmn. Exercises: Let U, { 1, 2, 3}, { 2}, { 2, 3, 4} [ 0,1] { x : 0 1} and D x. Which pairs f sets frm,,, D are disjint? WUT121 Lgic 156

10 4.9. Order f Operatins fr Set Operatrs. The rder f peratin fr set peratrs is as fllws: 1. Evaluate cmplement first 2. Evaluate and secnd. When bth are present, parenthesis may be needed, therwise wrk left t right. 3. Evaluate and third. When bth are present, parenthesis may be needed, therwise wrk left t right. Nte: Use f parenthesis will determine rder f peratins which ver ride the abve rder. Examples: Indicate the rder f peratins in the fllwing: {{ 1 2 { {{ { {{ { WUT121 Lgic 157

11 Exercises: Indicate the rder f peratins in the fllwing: {{ { {{ { { 123 {{ { Ntes. 1. and are peratins n sets, thus and can nly be put between tw sets. 2. and are peratins n statements, thus and can nly be placed between statements. Example: If,, and are sets then is interpreted as / WUT121 Lgic 158

12 WUT121 Lgic Set Laws Let,, and be subsets f a universal set U. That is,, U. Then fr all sets,, and fllwing set laws hld: 1. mmutative Laws: 2. ssciative Laws: 3. Distributive Laws: 4. Duble mplement Invlutin Law: 5. De Mrgan s Laws:

13 WUT121 Lgic Identity Laws: U 7. Negatin mplement Laws: U U U 8. Dminance Laws: U U 9. Idemptent Laws: 10. bsrptin Laws: 11. Set Difference

14 WUT121 Lgic Subset prperties f and 13. Subset prperty inclusin f intersectin 14. Subset prperty inclusin in unin 15. Transitive Prperty.