Answer:
a)z1 +z2 =z2 + z1 ...proved.
b) z1 + ( z2+ z3 )=(z1+z2)+z3 ... proved.
Step-by-step explanation:
It is given that there are three vectors z1 = a1 + ib1, z2 = a2 + ib2 and z3 = a3 + ib3. Now, we have to prove (a) z1 + z2 = z2 + z1 and (b) z1 + (z2 +z3) = (z1 + z2) + z3.
(a) z1 + z2 = (a1 +ib1) + (a2+ ib2) = (a1 +a2) + i(b1 +b2) {Adding the real and imaginary parts separately}
Again, z2 + z1 =(a2 +ib2) + (a1 +ib1) = (a2 +a1) + i(b2 +b1) {Adding the real and imaginary parts separately}
Hence, z1 +z2 =z2 + z1 {Since, (a1 +a2) = (a2 +a1) and (b1 +b2) = (b2 +b1)}
(b) z1 + ( z2+ z3 ) = [a1 + ib1] + [(a2 + a3 ) + i(b2 + b3 )] = ( a1 + a2 + a3) + i( b1+ b2+b3) {Adding the real and imaginary parts separately}
Again, (z1+z2)+z3 = [(a1+a2) +i(b1+b2)]+[a3+ib3] = ( a1 + a2 + a3) + i( b1+ b2+b3) {Adding the real and imaginary parts separately}
Hence, z1 + ( z2+ z3 )=(z1+z2)+z3 proved.
<em>First, you would multiply 2 x 16. </em>
2 x 16 = 32
<em>Using that number, subtract by 8. </em>
32 - 8 = 24
<em>Multiply 3 x 16. </em>
3 x 16 = 48
<em>Subtract the two numbers from step 2 and 3. </em>
48 - 24 = 24
F=2/3 of S
F+S=310
2/3 S + S = 310
(2S + 3S)/3=310
S=[(310)(3)]/5
S=186 students taking Spanish
F=2/3 of 186
F=124 students taking French
The y-intercept is where the curve crosses through the y axis (it intercepts it). In this case the y intercept would be worth three dollars at year 0. As years are time, and time is generally placed along the x axis.
Explanation:
a)
b) First, we have that
On the other hand,
Therefore, both expressions are equal. This makes sense, because selecting k elements from a group of n is the same than specify which elements you will not select. In order to specify those elements you need to select the n-k elements that will not be selected. Hence, each time you are selecting k from n, you are also selecting n-k from n, and from that reason both combinatorial numbers are equal.
