Recent content by HallsofIvy

Yes, you have shown that both are 'reflexive', mEm and mRm. An "equivalence relation" must also be symmetric. We need to show that "if mEn then nEm" and "if mRn the nRm". If mEn then 4 divides m- n so m- n= 4k for some integer k. Now n- m= -(m- n)= -4k= 4j where j=-k. If mRn 7m- 5n is even so...

Yes, that is correct

Of course, the empty set, $\phi$, is a subset of every set so it is true that $\phi\subset \{1, 2, 3\}$.

If tomorrow is Christmas then 2+ 2= 4.

This integral will be over the 6 faces of the cube. Do each separately. One face is z= 0 for x and y each from 0 to 1. On z= 0, the vector field zk= 0k so the integral is 0. That is also true on the faces x= 0 and y= 0. On z= 1 we integrate zk= 1k so we are integrating 1 for both x and y...

There is no "N" in the given equation so there can be no "N" in the answer! Have you left something out?

Well, it is knowing what these things are, what their definitions are! And note that it is "least common multiple", "lcm", NOT "lcd".

Okay, what IS an "orthogonal similarity transformation"? What is its definition? That's where I would start.

That looks straight forward given what you wrote. There is a 0.5% chance A will occur so a 1.000- 0.005= 0.995 or 99.5% none of A, or B, or C will happen. There is a 0.5% chance A WILL happen and then a 1- 0.02= 0.98 or 98% chance B will NOT happen. There is a (0.005)(0.98)= 0.0049 or 0.49%...

How are you DEFINING "vertical" in two dimensions? "Vertical" is a three dimensional notion depending on the direction of gravity. It might make sense to define "vertical", in a two dimensional sense, in the direction of the y-axis in an xy-coordinate system (but that is, of course, completely...

I would never say that a function is "trying" to do anything!

If you really intend $\int_0^{\pi/2} \sin(x^2) dx$, that cannot be integrated in terms of elementary function but can be approximated to any desired accuracy.

This is immediately impossible. f(2) cannot have two different values! We are given $y= f(x)= ax^2+ bx+ c$. The fact that (1, 5) is on its graph (a parabola) means that $y= 5= a(1)^2+ b(1)+ c= a+ b+ c$. The fact that (2, 10) is on the graph means that $y= 10= a(2)^2+ b(2)+ c= 4a+ 2b+ c$. If...

There is a constant acceleration, $-g= -9.8 m/s^2$ so after time t, the speed is $v(t)= v_0- 9.8t m/s$ where $v_0$ is the initial speed. The problem says that the velocity is $40- 10t$. I suspect that "10" is an approximation to g= 9.8 so that $v_0= 40 m/s$. The height after t seconds will be...

I would say that the first one is just drawn badly! They did NOT have a fractional number of players on the floor! There were 2+ 1+ 4+ 3= 10 players. For the next one, I have no idea what they mean by "shape"! I would probably say that this graph is shaped like a staircase!