Discrete Structure Chapter 2.1- Logic Circuits-29

8/10/2019 Discrete Structure Chapter 2.1- Logic Circuits-29

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Predicate Calculus

Predicate

Universal Quantifier

Existential Quantifier

De Morgans Laws

Other Rules for Quantifiers

Analogy Between Sets and Statements

SSK3003 Discrete Structures 1


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Predicate

An open sentence p(x) is a declarative sentence

that becomes a statement whenxis given a

particular value chosen from a universe of

values. An open sentence is also known as a

predicate.



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Example Predicate

Letp(x) = Ifx> 4, thenx+ 10 > 14 be an open sentence.

Letx U, where U= {1, 2, 3, 4, }. Find the truth value of

each statement formed when these values are substituted for

xinp(x).

p(1) is TRUE because 1 > 4 is FALSE.

p(2) is TRUE because 2 > 4 is FALSE.p(3) is TRUE because 3 > 4 is FALSE.



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Example Predicate (2)

p(x) = Ifx> 4, thenx+ 10 > 14

p(4) is TRUEbecause 4 > 4 is FALSE.

p(5) is TRUEbecause 5 > 4 and 5 + 10 > 14

are TRUE.

p(6) is TRUEbecause 6 > 4 and 6 + 10 > 14

are TRUE.

In fact,p(x) is TRUEfor all values ofxU.



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Universal Quantifier

The statement

For allx U,p(x)

is symbolized by

xUp(x).

The above statement is TRUE if and only ifp(x) is TRUE for every

x U.

The symbolis called the universal quantifier.



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Example: Universal Quantifier

Let U= {1, 2, 3, 4, 5, 6}. Determine the truth value of the

statement

xU[(x4)(x8) > 0].

Letp(x) = (x4)(x8) > 0.

p(1) is TRUE because (1 4)(18) > 0 is TRUE.





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Example: Universal Quantifier

(2)

p(x) = (x4)(x8) > 0.

p(4) is FALSEbecause (4 4)(48) > 0 is FALSE.

Therefore, the statement

xU[(x4)(x8) > 0]

is FALSEbecause 4 Uandp(4) is FALSE.



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Existential Quantifier

The statement

There exists anxUsuch thatp(x)

is symbolized by

xUp(x).

The above statement is TRUE if and only there is at

least one elementxUsuch thatp(x) is TRUE.The symbol is called the existential quantifier.



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Example: Existential Quantifier

Let U= {1, 2, 3, 4, 5, 6, 7, 8}. Determine the truth value of

xU[(x3)(x+ 2) = 0].

Letp(x) = (x3)(x+ 2) = 0.

p(1) is FALSE because (1 3)(1 + 2) = 0 is FALSE.

p(2) is FALSE because (2 3)(2 + 2) = 0 is FALSE.

p(3) is TRUE because (3 3)(3 + 2) = 0 is TRUE.



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Example: Existential Quantifier

(2)

Therefore, the statement

xU[(x3)(x+ 2) = 0]

is TRUEbecause we found 3 Usuch thatp(3) isTRUE.



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De Morgans Laws

The rules for the negation of quantified

statements are

~[xUp(x)][x U ~p(x)]

~[x Up(x)] [x U~p(x)].

These rules are called De Morgans laws.



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Example De Morgans Laws

Write the negation of

a) All university students like football.

b) There exists a university student whodoes not like football.

b) There is a mathematics textbook that is both

short and clear.

c) Every mathematics textbook is either not

short or not clear.



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Other Rules for Quantifiers

Other rules for statements containing

quantifiers are

~[xyp(x,y)]xy [~p(x,y)]~[xyp(x,y)]xy [~p(x,y)]

~[xyp(x,y)]x y[~p(x,y)]

~[xyp(x,y)]x y[~p(x,y)]

xyp(x,y)yxp(x,y)xyp(x,y)yxp(x,y)

xyp(x,y)yxp(x,y)



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Example: Other Rules for

Quantifiers

Let Ufor both variables be the nonnegative

integers 0, 1, 2, 3, 4, . Determine the truth

value of

a) xy[2x= y]

b) yx[2x= y]



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Example:Other Rules for

Quantifiers (a)

a)xy[2x= y]

b)The statement says that for every nonnegative

integerx, there is a nonnegative integer ysuchthat 2x= y. This is TRUEbecause, once having

chosen any nonnegative numberx, we can let y

be the double ofx.



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Example Other Rules for

Quantifiers (b)

b)yx[2x= y]

The statement says that there exists a

nonnegative integer ysuch that, for allnonnegative integersx, 2x= y. For it to be TRUE,

we would need to find a specific value of ythat

can be fixed and for which, no matter what

nonnegative integerxwe choose, 2x= y. Since

this is not possible, the statement is FALSE.



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Analogy Between Sets and

Statements

Letp(x) = xS, and q(x) = xT. Then


Union: x[xSTp(x) v q(x)]

Intersection: x[xSTp(x) ^ q(x)]

Complement:x[xS ~ p(x)]

Symmetric Difference:x[xSTp(x) q(x)]

Subset (ST)x[p(x) q(x)]

Equal (S= T)x[p(x) q(x)]


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Example: Sets and Statements

Let U= {1, 2, 3, 4, 5, 6}, let S= {x U:x< 3}, and let T= {x U:xdivides 6}.

Show S T.

We need to show that wherex S is TRUE then

x Tis TRUE.

By definition of S,x Sis TRUE if and only ifxis 1, 2, or 3.

However, 1, 2, and 3 all divide 6 sox Tis also TRUE.

Therefore, ST.



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Summary Section

Statements containing variables are called

predicatesand can be made into logical

statements with quantifiers. The quantifiers are

the symbols (for all) and (there exists).

These symbols refer to the particular universal

set for the variables in the predicate.



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Summary Section

Important rules of predicate calculus include

the following

~[xUp(x)] [x U ~p(x)]

~[xUp(x)][xU~p(x)]

x yp(x,y)y xp(x,y)

x yp(x,y)y xp(x,y)

x yp(x,y) y xp(x,y)



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Logic Circuits

Logic Circuits

NOT, AND and OR Gates

NAN and NOR Gates

XOR Gate



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Logic Circuits

Logic circuits can be found in computers,

telephones, digital clocks, and television sets

plus a great many more devices. In a logic circuitcurrent flows through gates to an output line.

The input current to the gate has only two states,

ON (1) or OFF (0). The output depends upon thetype of gate in the circuit.



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NOT, AND and OR Gates



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Example NOT, AND and OR Gates

Draw a logic circuit for three inputs,p, q, and r

and output (~p) (q r).

We will begin with the ANDgate.



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Example NOT, AND and OR Gates

(2)

Next we will add in the ORand NOTgates.



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Example Logical Statement (2)



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NAND and NOR Gates



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XOR Gate



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Summary Section

Logic circuits contain NOT, AND, OR, NAND, NOR

andXORgates.

Logic circuits frequently can be simplified

using the Table of Logical Equivalences.


Discrete Structure Chapter 2.1- Logic Circuits-29

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