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Sets and Logic…. Chapter 2

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1 Sets and Logic…. Chapter 2
An experiment….. Sets and Logic…. Chapter 2

2 Logical Laws DeMorgan’s Laws Commutative, Associative, and Idempotent
Β¬ π‘ƒβˆ§π‘„ 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ Β¬π‘ƒβˆ¨Β¬π‘„ Β¬ π‘ƒβˆ¨π‘„ 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ Β¬π‘ƒβˆ§Β¬π‘„ Commutative, Associative, and Idempotent Distributive π‘ƒβˆ§ π‘„βˆ¨π‘… 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ (π‘ƒβˆ§π‘„)∨(π‘ƒβˆ§π‘…) π‘ƒβˆ¨ Qβˆ§π‘… 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ π‘ƒβˆ¨π‘„ ∧ π‘ƒβˆ¨π‘… Absorption: π‘ƒβˆ¨ π‘ƒβˆ§π‘… 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ 𝑃 π‘ƒβˆ§ π‘ƒβˆ¨π‘… 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ 𝑃

3 Tautologies and Contradictions
Statements or Formulas which are always true are Tautologies Formulas that are always false are Contradictions π‘ƒβˆ§ π‘‘π‘Žπ‘’π‘‘π‘œπ‘™π‘œπ‘”π‘¦ ; π‘ƒβˆ¨ π‘‘π‘Žπ‘’π‘‘π‘Žπ‘œπ‘™π‘œπ‘”π‘¦ π‘ƒβˆ§ π‘π‘œπ‘›π‘‘π‘Ÿπ‘Žπ‘‘π‘–π‘π‘‘π‘–π‘œπ‘› ; π‘ƒβˆ¨(π‘π‘œπ‘›π‘‘π‘Ÿπ‘Žπ‘‘π‘–π‘π‘‘π‘–π‘œπ‘›)

4 Variables and Sets We will consider statements dependent upon a variable or a number of variables Ex’s P(x): x is a prime number D(x,y): x is divisible by y In this case we don’t have truth tables… we have truth sets

5 Conditional Statements
P implies Q: 𝑃→𝑄 If its raining and I don’t have my umbrella, then I’ll get wet. If Mary did her homework, then the teacher won’t collect it, and if she didn’t, the he’ll ask her to do it on the board

6 Equivalences 𝑃→𝑄 is equivalent to Β¬π‘ƒβˆ¨π‘„ 𝑃→𝑄 is equivalent to Β¬(π‘ƒβˆ§Β¬π‘„)

7 The Converse and Contrapositive
Consider 𝑃→𝑄 and 𝑄→𝑃 𝑄→𝑃 is called the converse of 𝑃→𝑄 ¬𝑄→¬𝑃 is the contrapositive of 𝑃→𝑄 The converse of 𝑃→𝑄 is not equivalent The contrapositive is…….. 𝑃→𝑄 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ ¬𝑄→¬𝑃

8 Chapter 2: Quantifiers Three ideas
For all or for every, we use the notation: βˆ€ There is one, or there exists: βˆƒ The universe……

9 Seven Operators Connectives Quantifiers
Β¬ ∧ ∨ β†’ ↔ Quantifiers βˆ€ π‘Žπ‘›π‘‘ βˆƒ This is really all we need to write any mathematical statement!

10 New Notation βˆ€π‘₯ π‘₯>2β†’ π‘₯ 2 >4 Is there and implied universe?

11 Examples βˆ€π‘₯ π‘₯ 2 >0 in what universe? βˆƒπ‘₯ π‘₯ 2 βˆ’2π‘₯+3=0 , Universe
βˆ€π‘₯ 𝑀 π‘₯ ∧𝐡 π‘₯ , where M(x) is ( x is a man) and B(x) is ( x has brown hair ) βˆ€π‘₯ 𝑀 π‘₯ →𝐡 π‘₯ βˆ€π‘₯𝐿(π‘₯,𝑦) where L(x,y) is a like function from x to y.

12 Write as Logical Statements
Someone didn’t do the homework Everything in that store is either overpriced or poorly made Nobody’s perfect Susan likes everyone who dislikes Joe π΄βŠ†π΅ π΄βˆ©π΅βŠ†π΅\C

13 Write as Statements Everybody in the dorm has a roommate he doesn’t like Nobody likes a sore loser Anyone who has a friend who has the measles will have to be quarantined If anyone in the dorm has a friend who has the measles, then everyone in the dorm will have to be quarantined If π΄βŠ†π΅, then A and C\B are disjoint

14 Multiple Quantifiers βˆ€π‘₯βˆƒπ‘¦ π‘₯<𝑦 βˆƒπ‘¦βˆ€π‘₯ π‘₯<𝑦 βˆƒπ‘₯βˆ€π‘¦ π‘₯<𝑦 βˆ€π‘¦βˆƒπ‘₯ π‘₯<𝑦
βˆƒπ‘₯βˆƒπ‘¦ π‘₯<𝑦 βˆ€π‘₯βˆ€π‘¦(π‘₯<𝑦)

15 EquivaleNce Involving Quantifiers
Quantifier Negation Laws Β¬βˆƒπ‘₯𝑃 π‘₯ 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œβˆ€π‘₯¬𝑃 π‘₯ Β¬βˆ€π‘₯𝑃 π‘₯ 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ βˆƒπ‘₯¬𝑃(π‘₯)

16 Examples Negate the statements, and then reexpress the results as equivalent positive statements π΄βŠ†π΅ Everyone has a relative he doesn’t like

17 Analyze the logical forms
All married couples have fights Everyone likes at least two people John likes exactly one person

18 Anayze the Logical Forms
Statements about the natural numbers β„• x is a perfect square x is a multiple of y x is prime x is the smallest number that is a multiple of both y and z

19 Analyze the statements
Statements about real numbers ℝ The identity element for addition is 0. Every real number has and additive inverse Negative numbers don’t have square roots Every positive number has exactly two square roots

20 2.3…. More on Sets Instead of just π‘₯ 𝑃(π‘₯) we can consider more general forms 𝑓(π‘₯) π‘₯∈[0,1] π‘Žπ‘₯+𝑏 𝑐π‘₯+𝑑 π‘₯πœ–[0,1]

21 More Examples π‘¦πœ– 3 π‘₯ π‘₯πœ–β„š π‘₯ 𝑖 π‘–πœ–πΌ βŠ†π΄ 𝑛 2 π‘›πœ–β„• and 𝑛 3 π‘›πœ–β„• are disjoint

22 The Power Set The Power Set β„˜ 𝐴 = 𝐡 π΅βŠ†π΄
β„˜ 𝐴 = 𝐡 π΅βŠ†π΄ Find the Power Set for 𝐴= π‘Ž,𝑏,𝑐

23 AnAlyze the logical forms
π‘₯πœ–β„˜ 𝐴 β„˜ 𝐴 βŠ†β„˜ 𝐡 π΅πœ– β„˜(𝐴) π΄βŠ†β„± π‘₯πœ–β„˜ 𝐴∩𝐡 π‘₯πœ–β„˜(𝐴)β‹‚β„˜(𝐡)

24 Intersections and Unions
β„±= 1,2,3,4 , 2,3,4,5}, 3,4,5,6 Find βˆ©β„±β€¦βˆͺβ„±

25 Definitions: Unions and Intersections
βˆ©β„±= π‘₯ βˆ€π΄πœ–β„± π‘₯πœ–π΄ = π‘₯ βˆ€π΄(π΄πœ–β„±β†’π‘₯πœ–π΄ βˆͺβ„±= π‘₯ βˆƒπ΄πœ–β„± π‘₯πœ–π΄ = π‘₯ βˆƒπ΄(π΄πœ–β„±βˆ§π‘₯πœ–π΄

26 Analyze the logical forms
π‘₯πœ–βˆ©β„± βˆ©β„±βŠˆβˆ©G π‘₯πœ–β„˜ βˆͺβ„± π‘₯πœ–βˆͺ β„˜(𝐴) π΄πœ–β„±

27 Another Example 𝐼= 1,2,3 𝐴 𝑖 = 𝑖,𝑖+1,𝑖+2,𝑖+3
𝐼= 1,2,3 𝐴 𝑖 = 𝑖,𝑖+1,𝑖+2,𝑖+3 Find βˆͺ π‘–πœ–πΌ 𝐴 𝑖 and ∩ π‘–πœ–πΌ 𝐴 𝑖

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