18)write the negation of the following statements
s: All students study mathematics at the elementary leve
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The negation of the given statement is ~s.thete exists a student who does not study mathematics at the elementary level.
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18)write the negation of the following statements s: All students study mathematics at the elementary level
17)write the negation of the following statements r: All birds have wings
Translate the following English sentences into statements of predicate calculus. a. All programmers enjoy discrete mathematics b. Some integers are not odd c. Every integer that is divisible by 2 is even d. There exists a natural number that is not a positive integer Refer to the statements of predicate calculus you provided for part 1. Write the negation of each of those statements. .
1. (2 points) Write a negation for the following statements: (a) If N is a nonagon, then N is a polygon. (b) If r is divisible by 56, then x is divisible by 7 and r is divisible by 8.
EXERCISES 6 Write and simplify the negation to the following le statements a) 3x<x2+1< 5 W . x>2 Vy<3 6) $9xty >3 .01.29 Izs1. (x-y< L da ih abzc ad bad c) 4x<1e y>2 acd d) a<b<ce bread 2 i) { x=1 Vy <3 e) x+1 <4 K x 2 <24 <3x+S Iz>y>x. f) ab. (cs d & cze) g) { x< 1 v { x=3 • lyce Tyzl
6. (4 marks) Write down the negation of each of the following statements. Then determine whether the statement or its negation is true, and explain why (a) x E R, y E R such that xy 5. (b) z, y E R+ such that V z E Z+, > z.
16)write the negation of the following statements q: There exists a rational number x such that x2=2
Discrete Math
in the final answer) 6. Write the negation of the following statements (without using (a). Un e Z+, 3p EP, n3 <p < (n + 1)3. (b). Vce R+,3K EZ+, Vn e Z, n > K → n n+100 < c.
15)write the negation of the following statements i) p: for every real number x,x2>x
Simplify the following sentences in predicate logic so that all the negation symbols are directly in front of a predicate. (For example, Vx ((-0(x)) + (-E(x))) is simplified, because the negation symbols are direct in front of the predicates O and E. However, Væ -(P(2) V E(x)) is not simplified.) (i) -(3x (P(x) 1 (E(x) + S(x)))) (ii) -(Vx (E(x) V (P(x) +-(Sy G(x, y))))) Write a sentence in predicate logic (using the same predicates as above) which is true...
Express each of the following using quantified state- ments over the domain S of all college students and the predicates C(x) meaning "x is a computer science major," and D(x) meaning "x takes discrete mathemat- ics." Which statements are equivalent to one another? Which statements are negations of one another? (d) Not every computer science major takes discrete mathematics. (f) You must take discrete mathematics if you are a computer science major (g) Some computer science majors do not take...