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Additional resources for (2,k)-Factor-Critical Graphs and Toughness
The universal set is shown in each case. Two sets are equal if and only if they have the same Venn diagram. In order to illustrate this, we again consider the distributive law DL1R ∪ (S ∩ T ) = (R ∪ S) ∩ (R ∪ T ). 1, and that for (R ∪ S) ∩ (R ∪ T ) is constructed in the lower half. The two are obviously identical. In the first part of this section, we applied the method of truth tables (developed for use with propositions) to set identities. We can also apply the methods of set 48 2 Sets and Data Structures theory to the analysis of propositions.
S ∪ T = S ∩ T . 13. S ∩ T = S ∪ T . 3 Proof Methods in Set Theory 51 14. (S ∩ T ) ⊆ S. 15. S ⊆ (S ∪ T ). 16. Use truth tables to represent the commutative and associative laws for ∪. 17. Use Venn diagrams to represent the commutative and associative laws for ∩. 18. For any sets R and S, prove R ∩ (R ∪ S) = R. 19. Prove, using Venn diagrams, that (R\S)\T = R\(S\T ) does not hold for all choices of sets R, S and T . 20. (i) Prove, without using truth tables or Venn diagrams, that union is not distributive over relative difference: in other words, prove that the following statement is not always true: (R\S) ∪ T = (R ∪ T )\(S ∪ T ).
Practice Exercise. Write the following numbers in floating point form, of length 4. 955. 114. 798. (4) 4/13. Simplified Floating Point Arithmetic To illustrate how floating point numbers are used in calculations, we shall use a system in which the mantissa is limited to four places. 113 × 102 . If two numbers have the same exponent, they are added by adding the mantissas. If the two exponents are different they must first be adjusted by increasing the smaller one. 21. 212 × 102 . Solution. 221 × 103 .