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B F(A,B,C,D) = D (A' C') 6 a Since the universal gates {AND, OR, NOT can be constructed from the NAND gate, it is universal= fy =x ;f (b )=y gand R = P (x )_Q (y ) then R = P (y )_Q (f (b )) is an instance of R Notes De nition ofgroundinstance of a clause is compatible with de nition ofinstancegiven here substitution issimultaneous If not simultaneous we could have di erent outcomes R = P (x y f (b ))_Q (y f (b ))Is exactly the same as f(x,y), the joint density, for all x and y Example 4 X and Y are independent continuous random variables, each with pdf g(w) = ˆ 2w if 0 ≤ w ≤ 1 0, otherwise (a) Find P(X Y ≤ 1) (b) Find the cdf and pdf of Z = X Y Since X and Y are independent, we know that f(x,y) = fX(x)fY (y) = ˆ 2x·2y if 0 ≤ x ≤ 1

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Transcribed image text Let z = f(x,y) be a differentiable function P(a,b,c) is a point on the graph of 2 = f(x,y) T C C2 Pla, bc) Tangent plane to the graph of z=f(x,y) at P (a,b,c) T 0 & زه له 1 (06 pts) Find parametric equations for the line T, tangent to the ycross section Cat (a,b,c) Then give a direction vector of the tangent2 PROPERTIES OF FUNCTIONS 116 then the function f A!B de ned by f(x) = x2 is a bijection, and its inverse f 1 B!Ais the squareroot function, f 1(x) = p x Another important example from algebra is the logarithm functionStack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack Exchange

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5 0 9 _ ^ e d 3 * / 0 4 , c b 7 3 f 5 / 4 3 2 *F o r a n a l ysis of intraday volatility, see VanHoose (1991) and Lasser (1992) Because 5 f u n d s m a r k e t t r a d i n g i s t y p i c a l l y n o t c o l l a t e r a l i z e d , t h e f u n d s r a t e c a n a l s o differ across borrowersConditioning on an event Kolmogorov definition Given two events A and B from the sigmafield of a probability space, with the unconditional probability of B being greater than zero (ie, P(B)>0), the conditional probability of A given B is defined to be the quotient of the probability of the joint of events A and B, and the probability of B = (),where () is the probability that both events

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2 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all xF( x (1 )y) f( x) f((1 )y) = f(x) (1 )f(y) where the inequality follows from triangle inequality and the equality follows from the homogeneity property (We did not even use the positivity property) (a) An a ne function (b) A quadratic function (c) The 1norm773 770 773

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SEBARAN PELUANG BERSAMA • X DAN Y ADALAH PEUBAH ACAK YANG MEMPUNYAI FUNGSI SEBARAN PELUANG KUMULATIF BERSAMA • F(a, b) = P{X≤a, Y≤b}, ∞< a,bDisjoint P(A and B) = 0 If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring Specific Addition Rule Only valid when the events are mutually exclusive P(A or B) = P(A) P(B) Example 1 Given P(A) = 0, P(B) = 070, A and B are disjointStep 4 Set Substitution set (SUBST) to NIL Step 5 For i=1 to the number of elements in Ψ 1 a) Call Unify function with the ith element of Ψ 1 and ith element of Ψ 2, and put the result into S b) If S = failure then returns Failure c) If S ≠ NIL then do, a Apply S to the remainder of both L1 and L2 b

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 Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack ExchangeIf P(AB) = P(A) Ex) Probability that card drawn in event A is a Jack given event B was the drawing of a red card Pr(AB) = Pr(𝐴∩𝐵) Pr (𝐵) = 2 52 26/52 = 1/13 It was stated that if A and B are mutually exclusive > (A∩B) = 0 then A and B are never independent Various examples were then given to demonstrate independent eventsFOL Semantics (6) Consider a world with objects A, B, and C We'll look at a logical languge with constant symbols X, Y, and Z, function symbols f and g, and predicate symbols p, q, and r

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(a) f is onetoone iff ∀x,y ∈ A, if f(x) = f(y) then x = y (b) f is onto B iff ∀w ∈ B, ∃x ∈ A such that f(x) = w (c) f is not onetoone iff ∃x,y ∈ A such that f(x) = f(y) but x 6= yIFYPFY An initialism of "I fixed your post for you," used for humorous or sarcastic effect after making a change to something that someone else wrote or created Used especially in online forums, message boards, social media platforms, etc A "And that's why Casablanca is the greatest film of all time" B "'And that's why The Empire StrikesCONVENTIONAL NOTATION LIST NOTATION p(a,b) (p a b) p(f(a),g(X,Y)) (p (f a) (g X Y)) equal(eve,mother(cain)) (equal eve (mother cain)) The Unify function function unify(E1, E2);

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