EXERCISES 6 Write and simplify the negation to the following le statements a) 3x<x2+1< 5 W...
16)write the negation of the following statements q: There exists a rational number x such that x2=2
2. Let D-E-(-2,-10,1,2). write negations for each of the following statements and determine which is true, the original or the negation. Vx e D,3y E E such that xy 2 y True: OriginalNesgation a. b. 3x E D such that Vy E E, x y True: Original Negation
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.
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...
4. Simplify and state the restrictions. 2x+8 4x+16 a) 3x 6x2–5x+1 b) x2-4 Х x2-x-2 x2-3x 2x2 4 1 c) x2+3x+2 + 1 x2+4x+3 11x d)- x2+3x-28 X-4
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.
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.
7. Determine the truth value of the following statements. (a) Bx E Q, x2-2 = 0 (b) BxE Z,Vy e R, x +y = 1 (c) V e Z, Fy e R, x+y = 1 (d) 3x E Z,By e Z, x2 + y2 = 9
1. Evaluate the iterated integrals: x2+2x+y a. JR 3x+3y dA, R: 15x32,0 sys 1 (Hint: Simplify the integrand first.) b. S ey/*dA where R is the region in the xy-plane bounded between y = x2 and y = x over the interval 1sx52. c. So Sex Sx**2 x dydzdx
2. Find the derivative and simplify. y = (3x-4)2 (1-x²)6