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MAT 1348 Study Guide - Final Guide: Equivalence Class

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17 May 2020

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University of Ottawa

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When de ning a function we require that the function takes each ele- ment from the domain to exactly one element in the codomain. Given a function f : a b, one can associate it to a subset gf of. A b, where elements are some ordered pairs (a, b) (a a, b b). Note that since f is a function from a to b, for each x a, there is exactly one (x, y) gf. Now we introduce a new notion, called a relation, which will serve as a generalization of the notion of function: Defn: a binary relation r from a set a to a set b is a subset of a b. We write (a, b) r (or arb) to denote that a is related to by r . Fact: for nite sets a and b, the number of relations from a to b is |p(a b)| = 2|a b| = 2|a||b|

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Related Questions

I have the answer for the following question. But some of theanswers are not correct.

Please do the following question by yourself and check which arethe wrong answers,please post your answers and compare to it.

Let r in and u =(1 1 0), v = (1 4 1), w = (r2 1 r2), b = (3+2r 5+12r 2r). For which values r in is the set {u. v, w} linearly independent? For which values r in is the vector b a linear combination of u. v and w? For which of these values of r can b be written as a linear combination of u. v and w in more than one way? The set 3 of all column vectors of length three, with real entries, is a vector space. Is the subset B = {(x y z) in 3 | xy + yz = 0} a subspace of 3? Justify your answer. Show that the set of all twice differentiable functions f : satisfying the differential equation sin(x) f"(x) + x2f(x) = 0 is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f. Let S be the following subset of the vector space 3 of all real polynomials p of degree at most 3: S = {p in 3 | p(l) = 0, p'(l) = 0}, where p' is the derivative of p. Determine whether S is a subspace of 3. Determine whether the polynomial q(x) = x - 2x2 + x3 is an element of S. u = (1 1 0), v = (1 4 1), w = (r2 1 r2) if {u, v, w} is linearly independent set 1(4r2 - 1)-1(r2 - r2) ne 0 4r2 ne 1 r ne plusmn1/2 this set is linearly independent set for values of e ne plusmn1/2 - - - - - - - (3+2r 5+12r 2r) = a1 (1 1 0) + a2 (1 4 1) + a3 (r2 1 r2) this forms system of equations in a1, a2, a3 if 4r2 - 1 ne 0 then system has unique solution if 4r2 - 1 = 0 and -2 - 4r = 0 rArr r = -1/2, the system has more than one solution if 4r2 - 1 ne 0 b can be written as linear combination of u,v,w as uniquely if r = -1/2 then b can be written as linear combination of u,v,w more than one way - - - - - - - - - (1 1 -1) in B and (1 0 1) in B (1 1 -1) + (1 0 1) = (2 1 0) not in B since 2times1+1times0 ne 0 so addition of vector is failed, this set is not forms subspace consider the following differential equation (sinx)f"(x) + x2f (x) = 0 each solution is continuous function the set of all solutions is subset of space of set of all continuous functions n from R to R let f,g are two solutions (sinx)f"(x) + x2f (x) = 0 (sinx)g"(x)+x2g(x) = 0 ccheck whether af+bg is solution or not (sinx)f"(x) + x2f(x) = 0 rArr (sinx) af"(x) + x2?f (x) = 0 (sinx)g"(x) + x2g (x) = 0 rArr (sinx)bg" (x) + x2bg (x) = 0 (sinx)(af"(x)+bg"(x)) +x2 (af (x) + bg(x)) = 0 af+bg is also solution to differential equation the set of all solutions to differential equation forms subspace - - - - - - - - - - consider the following subset of P3 {p in P3/.p(l)=0, p'(l) = 0} let p, q in S rArr p(l)= 0, p'(l) = 0 and q(l) = 0, q'(l) = 0 (ap + bq) (l) = ap (l) +b q(l) = a (0) + b (0) = 0 (ap + bq)'(1) = ap'(l) +bq'(1) = a (0) + b(0) = 0 p, q in S rArr ap +bq in S then S is subspace q(x) = x - 2x2 + x3 q(1) = 1 - 2 + 1 = 0 q'(x)=1 - 4x + 3x2 q'(l) = 1 - 4 + 3 = 0 q is element of S

Can some do this ?

Complete this assignment in Maple. When you are finished, save it to your hard drive and email a copy of your Maple worksheet to me at Write "Math 1204 Lab 4: Name1 and Name2" exactly like this as the subject of your email, where Name1 and Name2 are your names (if you work on this alone, put one name). This lab will get you to construct an interpolating polynomial from given data. Open Maple and load the Linear Algebra package as usual. A chemical company is testing the exothermic properties of a new molecule being combined with 100 ml Hydrochloric Acid at 21 degrees C. Experimentally, they have recovered the following data: Look in your notes for Section 2.7 and remind yourself how polynomial interpolation works. Using MAPLE and our techniques from Section 2.7, find an interpolating polynomial that passes through all six data points. To do this, it might be easier to set each amount of molecule as a variable, like this: and so on. Using this polynomial estimate the temperature of the HCl when 400 mg of the molecule is added 125 mg is added. 50 mg is added. We will now graph the data points and your polynomial on the same graph. First, we need to include the 'plots' package from Maple with the following command: > with (plots); Make a list of your ordered pairs of data and call it K. Example: if I wanted to store the ordered pairs (1, 2) and (6, -1) in a list named K, I would input the following command: > K: [[1, 1], [6, -1]]; Note the MAPLE uses square brackets for ordered pairs, and lists as well. Plot your points with the following command: > pointplot(K); Input your interpolating polynomial into MAPLE and call it f. Plot your interpolating polynomial with the following command: > plot(f, x = 1..6); Note that the second part of the plot function tells MAPLE the bounds for x in our graph. Plot these together on the plot function tells MAPLE the bounds for x in our graph. Plot these together on the same graph in the following way: > display ({pointplote(K), plot(f,x=-1..6)}); What do you notice about your data points in relation to your function? An intern is asked to run the experiment again. They recover the following data. Attempt to use polynomial interpolation for this set of data. Why doesn't work? Explain why there is no function that can model this data.