1. IB Math Studies – Topic 3

2. IB Course Guide Description

4. Set Theories A set is any collection of things with a common property: it can be finite. ▫Example: set of students in a class If A= {1,2,3,4,5} then A is a set that contains those numbers

5. Subsets

6. Union and Intersection

7. Venn Diagrams Venn Diagrams are diagrams used to represent sets of objects, numbers or things. The universal set is usually represented by a rectangle whereas sets within it are usually represented by circles or ellipses.

8. Sets within Venn Diagram Disjoint or Mutually Exclusive sets

9. Logic Proposition ▫The building block of logic. ▫This is a statement that can have one of the two value, true or false. Negation The negation of a proposition is formed by putting words such as “not” or “do not.” The negation of a proposition p is “not p” and is written as ¬p.  For example:  p: It is Monday.  ¬p: It is not Monday.

10. Truth Tables A truth table shows how the values of a set of propositions affect the values of other propositions. A truth table for negation p¬ p TF FT

11. Compound Propositions

12. Conjunction/Disjunction and Venn Diagrams

13. Tautology A tautology is a compound proposition that is always true, whatever the values of the original propositions. Example: When all the final entries are ‘T’ the proposition is a Tautology. p ¬p T F T F T T

14. Contradiction A Contradiction is a compound proposition that is always false regardless of the truth values of the individual propositions. Example: When all the final entries are ‘F’ the proposition is a Contradiction. p¬p TFF FTF

15. Logically Equivalent pq TTTF FTFT TFFT FFTF ¬p¬q FFF TFT FTT TTF

16. Implication If two propositions can be linked with “If…, then…”, then we have an implication. p is the antecedent and q is the consequent The symbol would be  For example: ▫ P: You steal ▫ Q: you go to prison Therefore, the words “If” and “then” are added. “if you steal, then you go to prison.”

17. Converse, Inverse, and Contrapositive Converse: q  p Inverse:  p   q Contrapositive:  q   p ▫ For example: ▫ P: It is raining ▫ Q: I will get wet  Converse: “If it is raining, then I will get wet.”  Inverse: “It it isn’t raining, then I won’t get wet.”  Contrapositive: “It I’m not wet, then it isn’t raining.”

18. Probability Probability is the study of the chance of events happening

19. Combined Events

20. Sample Space There are various ways to illustrate sample spaces: Sample space of possible outcomes of tossing a coin. Listing Sample space = {H,T} 2-D Grids Tree Diagram

21. Theoretical Probability A measure of the chance of that event occurring in any trial of the experiment. The formula is:

22. Using Tree Diagrams

23. Tree Diagrams – Sampling Sampling is the process of selecting an object from a large group of objects and inspecting it, nothing some features The object is either put back (sampling with replacement) or put to one side (sampling without replacement).

24. Laws of Probability TypeDefinitionFormula Mutually Exclusive Events Events that cannot happen at the same time P(A ∩ B) = 0 P(A  B) = P(A) + P(B) Combined Events (a.k.a Addition Law) Events that can happen at the same time P(A  B) = P(A) + P(B) – P(A∩B) Conditional ProbabilityThe probability of an even A occurring, given that event B occurred. P (A | B) = P (A ∩ B) P (B) Independent EventsOccurrence of one event does NOT affect the occurrence of the other P(A ∩ B) = P(A) P(B)