document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205232>Question 1205232</A>: <I><SPAN STYLE=\"word-break: break-all;\"> for the relations: A x B, such that:<BR>\n" );
document.write( "A = {-7,-6,..., -4} and B = {42,43,...,45}\r<BR>\n" );
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document.write( "count how many possible ways N1 to have functions p = A --> B\r<BR>\n" );
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document.write( "N1 = \r<BR>\n" );
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document.write( "count how many possible ways N2 to have p as one to one function.<BR>\n" );
document.write( "N2= \r<BR>\n" );
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document.write( "count how many possible ways N3 to have p NOT onto.<BR>\n" );
document.write( "N3 = <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841892 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=841892\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "A = {-7, -6, -5, -4}<BR>\n" );
document.write( "B = {42, 43, 44, 45}<BR>\n" );
document.write( "Each set has 4 elements.<BR>\n" );
document.write( "It's not clear why your teacher used the triple dot notation when it's not needed.\r<BR>\n" );
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document.write( "For the input x = -7, from set A, there are 4 choices it could map to in set B. Same with x = -6 and so on.<BR>\n" );
document.write( "There are 4*4*4*4 = 4^4 = <font color=red>256</font> different functions possible.<BR>\n" );
document.write( "A function is a collection of (x,y) points where x doesn't repeat itself. <BR>\n" );
document.write( "The y values can repeat, but the function wouldn't be one-to-one.<BR>\n" );
document.write( "Often the (x,y) points are connected with some kind of line or curve to form a graph; however, in this case, the points will be separate disconnected islands.\r<BR>\n" );
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document.write( "A function is considered one-to-one (aka injective) when we don't have repeated outputs. This is when each input is assigned a unique output. <BR>\n" );
document.write( "You can think of it like pairing up dance partners.<BR>\n" );
document.write( "The input x = -7 has 4 choices to pick from.<BR>\n" );
document.write( "After we pick something from B, there are 4-1 = 3 choices left for the next input. Then 3-1 = 2 choices for the third input, and finally 1 choice for the last input.<BR>\n" );
document.write( "There are 4*3*2*1 = <font color=red>24</font> one-to-one functions possible.\r<BR>\n" );
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document.write( "n(A) = number of values in set A<BR>\n" );
document.write( "Because n(A) = n(B), a function that is onto is automatically one-to-one, and vice versa.<BR>\n" );
document.write( "The proof is left as an exercise for the student.\r<BR>\n" );
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document.write( "\"Onto\" is the same as \"surjective\". They both refer to the idea that every item in set B is targeted.\r<BR>\n" );
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document.write( "Here's an example of a surjective function<BR>\n" );
document.write( "{(-7,42),(-6,43),(-5,44),(-4,45)}<BR>\n" );
document.write( "and here's something that isn't surjective.<BR>\n" );
document.write( "{(-7,42),(-6,42),(-5,44),(-4,45)}<BR>\n" );
document.write( "in the 2nd example, the output y = 43 isn't targeted in set B. <BR>\n" );
document.write( "Therefore, the 2nd example isn't surjective. Notice how the 2nd example isn't injective either since we have a repeated y value.\r<BR>\n" );
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document.write( "The term bijective means \"both injective and surjective\".\r<BR>\n" );
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document.write( "Here's an article that talks about the terms mentioned<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.mathsisfun.com/sets/injective-surjective-bijective.html\">https://www.mathsisfun.com/sets/injective-surjective-bijective.html</a>\r<BR>\n" );
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document.write( "We found there are 24 bijective functions and 256 functions possible.<BR>\n" );
document.write( "That must mean there are 256-24 = <font color=red>232</font> functions that are not surjective. \r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "N1 = <font color=red>256</font><BR>\n" );
document.write( "N2 = <font color=red>24</font><BR>\n" );
document.write( "N3 = <font color=red>232</font><BR>\n" );
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