Dr. Joanna Kosmala-Anderson Prof. Louise M. Wallace National Breastfeeding Training Needs Survey
Topic 3 Sets, Logic and Probability Joanna Livinalli and Evelyn Anderson.
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Transcript of Topic 3 Sets, Logic and Probability Joanna Livinalli and Evelyn Anderson.
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Topic 3 Sets, Logic and Probability
Joanna Livinalli and Evelyn Anderson
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IB Course Guide description
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Topic 3.1 and 3.3: Set Theory
Set Notation
Symbol Definition Means…
U universal set All the elements given in a question
is an element of Is an element or member of the set
is not an element of Is not an element or member of the set
‘ Complement opposite of the set
the empty set an empty set with no members/elements
Intersection An overlap; only the elements in both sets.
Union A marriage; all the elements in each set, including all elements that are in both.
Subset A smaller set contained wholly within a larger set
n(A) Number of elements inside set A
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Topic 3.2:Venn diagrams and sets
Intersection
Union Subset
Mutually exclusive events
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(A B)’
A’ B
(A B)’
A’ B
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Example Problem A group of 40 IB students were surveyed about the languages they have chosen at IB: E = English, F = French, S = Spanish.
• 3 students did not study any of the languages above.• 2 students study all three languages• 8 study English and French• 10 study English and Spanish• 6 study French and Spanish• 13 students study French• 28 students study English
(a) Draw a Venn diagram to illustrate the data above. On your diagram write the number in each set.
(b) How many students study only Spanish?
(c) On your diagram shade (E F)’ , the students who do not study English or French.
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Check your answers (a)
(b) There 4 students who do Spanish only (c) Shade everything but the union of E and F
(where the 6 is)
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Topic 3.4 – 3.7 : Logic and truth tables
• Propositions – A statement that can be either true or false. For example, the baby is a girl.
• Negation – A proposition that has become negative: For example, the baby is not Australian. Be careful that it is a negative and not the opposite. The symbol used is ¬.
• Conjunction – the word ‘and’ used to join two conjunctions together. For example, the baby is a girl and Australian. The symbol used is .
p ¬p
T F
F T
p q p q
T T T
T F F
F T F
F F F
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• Disjunction – the word ‘or’ used to join the conjunctions together. For example, the baby is a girl or Australian. The symbol used is .
• Exclusive Disjunction - is true when only one of the propositions is true. The word ‘or’ is used again, but in this case you must write “or, but not both”. The symbol used is
p q p q
T T T
T F T
F T T
F F F
p q p q
T T F
T F T
F T T
F F F
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• Implication – Using the words if …… then with two propositions. For example: If you do not sleep tonight then you will be tired tomorrow. The symbol used for implication is .
• Equivalence – If two propositions are linked with “…if and only if…”. Then it is an equivalence. p q is the same as p q and q p. Or using notation: (p q) = (p q) (q p)
p q pqT T T
T F F
F T T
F F T
p q pq
T T T
T F F
F T F
F F T
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p: My shoes are too small q: My feet hurt – Implication: p q. If my shoes are too small,
then my feet hurt. – Converse: q p. If my feet hurt, then my shoes
are too small. – Inverse: p q. If my shoes are not too small,
then my feet do not hurt. – Contrapositive: q p. If my feet do not hurt,
then my shoes are not too small.
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• Tautology – a statement that produces True (T) throughout the column of the truth table.
Column (?) would be a tautology
• Contradiction – a statement that produces False (F) throughout the column of a truth table.
Column (?) Would be a contradiction
p q ?T T T
T F T
F T T
F F T
p q ?T T F
T F F
F T F
F F F
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Example Problem
P Q r p q (p q) (p q) rT T T T T F T F T T F F F T T F T F F F T F F F
p: Andrea studies IB English.q: Andrea studies IB Spanish.r: The school offers at least 2 IB languages.
(a) Write the following in logical form. If Andrea studies English and Spanish, then the school offers at least 2 languages.
(b) Write the following statement in words: ¬p ¬q
(c) Copy and complete the truth table below.
(d) Is (p q) r a tautology, contradiction or neither?
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Check your answers (a) (p q) r
(b) If Andrea does not study English then she will not study Spanish
(c)
(d) Neither a tautology nor a contradiction
p q r p q (p q) (p q) rT T T T F TT T F T F TT F T F T TT F F F T FF T T F T TF T F F T FF F T F T TF F F F T F
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Probability
Probability ranges between 0 and 1
or
0 and 100%
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Terminology
• Trials—times experiment is repeated• Outcomes—different possible results• Frequency—the number of times an outcome
is repeated• Relative frequency—the number of times an
outcome is repeated in terms of a fraction or percentage
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Example Problem
• In assigning 32 assignments, Ms. Boswell has noticed that Maxine has done her homework 12 times.
• Describe the trials, outcomes, frequency, and relative frequency of this situation.
Note: The greater the number of trials, the more reliable the information becomes, and the closer the relative frequency will be as a predictor for future outcomes.
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3.8 Equally Likely Events
• If the probability of an event A given by P(A)=n(a)/n(U)
• Probability of a complementary event isP(A’)=1-P(A)
• Example: In a coin, there are two possible outcomes—heads or tails. So the probability of it not being tails is equal to 1-(the probability of it being heads, or .5), which is also .5.
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Venn DiagramsThere are 48 students. 22 watch Dragon Ball Z, 15 watch Pokémon, 20 watch Naruto, and 12 watch none.
5
3
What is the probability of (DUP)’? DUP=(30/48)=(5/8)(DUP)’=1-(5/8) or (3/8)What is the probability that the student watches Dragon Ball Z? (22/48) or (11/22)What is the probability the student watches Pokémon and Naruto, but not Dragon Ball Z? (3/48)What is the probability the student watches Naruto or Pokemon, but not both? ((7+6)+(3+5))/48 or 21/48=(7/16)
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What is the probability the coin will fall on heads three times in a row? (Tree Diagrams)
Heads (1/2)
Tails (1/2)
Tails (1/2)
Heads (1/2)
Tails (1/2)
Heads (1/2)
(1/2)*(1/2)*(1/2)=(1/8)
Number of possibilities for a particular outcome/Number of total possible outcomes
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If there are 5 red marbles and 2 blue marbles in a bag, what is the probability that when
picking one out, one will be blue and the other red? (the marbles are not replaced when
removed)
Red (5/7)
Blue (2/7)
Blue (2/6)
Red (4/6)
Blue (1/6)
Red (5/6)
((5/7)*(2/6))+((2/7)+(5/6))=(10/21)
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Table of Outcomes
• Example: Rolling two pairs of dice
http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i10/bk8_10i3.htm
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Mutually Exclusive Events
• A situation where only one outcome is possible (i.e., if the coin is heads, it cannot be tails also.)
• A die is tossed. What is the probability the shown is 4?
• What is the probability that it is 4 or 3?• What is the probability that it is 4 and 3?• What is the probability that the dice will roll a 4,
and when tossing a coin at the same time, it will be heads?