02 more on sets

Sets

Exercises

[True/False] (1 pt each)5. A null set is a proper subset of itself.6. The set of counting numbers less than

100 billion is an infinite set.7. All equivalent sets are equal sets.

3Math 11 College Algebra

Exercise

Shade the area which represents the ff set: (A U B) – C

4Math 11 College Algebra

U

A B C

The more you READ, The more you KNOW

The more you KNOW, The smarter you GROW

The smarter you GROW, the stronger your VOICE

When speaking your MIND or making a CHOICE

Power set - subsets of a given set- denoted by P (A)- examples, A = {1, 2, 3}P (A) = { , {1}, {2}, {3}, {1, 2}, {1,3}, {2, 3}, {1, 2, 3} }

P ({ }) =

Math 11 College Algebra 7

{ , { }}

Cardinality of a Power set - n(P (A)) or |P (A)| = 2 n(A) - examples, A = {1, 2, 3}P (A) = { , {1}, {2}, {3}, {1, 2}, {1,3}, {2, 3}, {1, 2, 3} }

n(P (A)) = 23 = 8P ({ }) =

n(P (A)) = 21 = 2

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{ , { }}

Other properties of Set Operations• Distributive Property A (B C) = (A B) (A C)

A (B C) = (A B) (A C)Examples:

1) Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}. C= {4, 5}.

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Other properties of Set Operations• De Morgan’s Law (A B)’ = A’ B’

(A B)’ = A’ B’Examples:

1) Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}. 

Show that (A B)∪  ' = A ' ∩ B'.

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Solution : 

U = {1, 2, 3, 4, 5, 6} A = {2, 3}

B = {3, 4, 5} 

A B = {2, 3} {3, 4, 5} ∪ ∪= {2, 3, 4, 5}

(A B)∴ ∪  ' = {1, 6}

Also A ' = {1, 4, 5, 6}B ' = {1, 2, 6}

A∴ ' ∩ B' = {1, 4, 5, 6} ∩ {1, 2, 6} = {1, 6}

Hence (A B)∪ ' = A ' ∩ B'

• If ξ = {a,b,c,d,e}, A = { a,b,d} and B = {b,d,e}. Prove De Morgan's

Applications

• Solving word problems using Venn Diagram

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Applications

• Draw the Venn diagram of the ff sets:Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} A = {0, 1, 2, 3} B = {2, 3, 4, 5} C = {3, 4, 5, 6, 7}

16 June 2011Math 11 College Algebra 14

2. From the data compiled by the University Registrar, it has been learned that 200 students have enrolled in Math, 180 in History, while 170 have enrolled in English. Forty- five have enrolled in both History and English, 50 in Math and English and 40 in both Math and History. Fifteen students, meanwhile, enrolled in all three subjects. Out of all 500 students, how many enrolled...a.) in Math only?b.) in both Math and English but not History?c.) in at least one of these courses?d.) in none of these courses?

15Math 11 College Algebra

1. In a survey, 150 people were selected and asked to write down social network accounts. The result was:

70 people have Facebook62 people have Twitter

50 Instagram 44 have FB & Twitter 20 chose Twitter & Instagram 26 chose FB & Instagram

10 chose FB, Twitter, & Instagram

16Math 11 College Algebra

1. In a survey, 150 people were selected and asked to write down their favorite entertainment places in Davao City. The result was:

Find out:a) How many didn’t have a social network account?b) How many people have FB only?c) How many people have Twitter only?d) How many people have Instagram only?e) How many have social network accounts but no FB?f) Who did not have Twitter?

17Math 11 College Algebra

Find out:a) How many didn’t have a social network account?b) How many people have FB only?c) How many people have Twitter only?d) How many people have Instagram only?e) How many have social network accounts but no FB?f) Who did not have Twitter?

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