CD_ONLY_IBHM_Ch00.pdf

CD_ONLY_IBHM_Ch00.pdf - 0 Presumed Knowledge 0.1 Sets...

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1 0.1 Sets Number S ystems The number systems that it is important to be aware of are as follows: The natural numbers, denoted by are The integers, denoted by are The positive integers, denoted by The rational numbers, denoted by are defined as the set of numbers that can be expressed as a fraction of two integers, i.e. The real numbers, denoted by can be considered as any number on the number line including irrational numbers such as 2 2 , p . , b x : x a b , a , b H , b 0 r . , , are 5 1, 2, 3, 4, p 6 . 5 p , 3, 2, 1, 0, 1, 2, 3, 4, p 6 . , 5 0, 1, 2, 3, 4, p 6 . , 0 Presumed Knowledge These brackets denote a set. denotes the empty set. means “is a member of”. e denotes the universal set, which means the set of numbers upon which other sets are defined. denotes the intersection of two sets. Intersection means the members of the two sets that are elements in both sets. This is analogous to “and”. ¨ H 56 Set notation Within this syllabus, set notation will be used. A set is a collection of objects; for this syllabus sets tend to be collections of numbers. 4 5 2 3 1 0 1 2 3 4 The diagram below shows the relationship between these sets:
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0 Presumed Knowledge 3 Example This can be expressed as the set This means the set of numbers x , where x is greater than or equal to 1, where x is a member of the real numbers. 5 x : x 1, x H 6 . 4 5 2 3 1 0 1 Example Draw and on a Venn diagram and write down the intersection and union of P and Q . State why P is a subset of Q . and Here all of set P is contained in set Q and so P is a subset of Q , that is, P ( Q . P ´ Q 5 2, 3, 5, 7 6 . P ¨ Q 5 3, 5 6 Q 5 2, 3, 5, 7 6 P 5 3, 5 6 Q P 2 7 5 3 It is important to be able to express an inequality as a set. 1 Express each inequality as a set. a b c d 2 For the sets given, draw a Venn diagram and state the intersection and union of the sets. a b c d e 3 If and state the set A . Draw a Venn diagram of these two sets. A ¨ B 5 7, 9, 11 6 , B 5 3, 7, 9, 11, 13 6 , A ( B P 5 5, 1, 2 6 , Q 5 5, 1, 2, 7, 11 6 K b 1 2 , 3 4 , 9 r , L b 1 2 , 9, 11, 17 r F 5 3, 2, 10 6 , G 5 1, 2, 7 6 C 5 0, 3, 7 6 , D 5 3, 9, 11, 13 6 A 5 1, 2, 3 6 , B 5 3, 4, 5 6 Exercise 1 2 3 1 0 1 2 3 2 3 4 1 0 1 2 2 3 4 1 0 1 2 2 3 1 0 1 2 0 Presumed Knowledge 2 Example Draw and on a Venn diagram and write down the intersection and union of A and B . The sets A and B overlap as an intersection exists. The intersection is The union of the sets is A ´ B 5 1, 2, 5, 7, 9 6 . A ¨ B 5 2, 5 6 . B 5 2, 5, 7, 9 6 A 5 1, 2, 5 6 Example Draw and on a Venn diagram and write down the intersection and union of C and D . Here the sets do not overlap so C ´ D 5 2, 3, 7, 11, 13, 19 6 C ¨ ˛˛ D . D 5 2, 3, 13 6 C 5 7, 11, 19 6 A B 7 2 1 5 9 C D 19 7 11 13 2 3 denotes the union of two sets. Union means all the members that are in either set. This is analogous to “or”. or means “is a subset of”. This is analogous to and for numbers. 6 ( ´ Example What does mean?
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