document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191688>Question 1191688</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a set S = ℝ, prove that a relation R on S where (a , b) ∈ R and a - b = 0 is an equivalence relation </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849102 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849102\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To prove that the relation R on S = ℝ (real numbers), where (a, b) ∈ R if and only if a - b = 0, is an equivalence relation, we need to show that R is reflexive, symmetric, and transitive.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Reflexivity:**\r<BR>\n" );
document.write( "\n" );
document.write( "We need to show that for all a ∈ S, (a, a) ∈ R.\r<BR>\n" );
document.write( "\n" );
document.write( "If a ∈ S, then a - a = 0.  Since a - a = 0, (a, a) ∈ R by the definition of R.  Therefore, R is reflexive.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Symmetry:**\r<BR>\n" );
document.write( "\n" );
document.write( "We need to show that for all a, b ∈ S, if (a, b) ∈ R, then (b, a) ∈ R.\r<BR>\n" );
document.write( "\n" );
document.write( "Suppose (a, b) ∈ R.  This means that a - b = 0.<BR>\n" );
document.write( "If a - b = 0, then we can multiply both sides by -1 to get -(a - b) = -0, which simplifies to b - a = 0.<BR>\n" );
document.write( "Since b - a = 0, (b, a) ∈ R by the definition of R.  Therefore, R is symmetric.\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Transitivity:**\r<BR>\n" );
document.write( "\n" );
document.write( "We need to show that for all a, b, c ∈ S, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.\r<BR>\n" );
document.write( "\n" );
document.write( "Suppose (a, b) ∈ R and (b, c) ∈ R.<BR>\n" );
document.write( "This means that a - b = 0 and b - c = 0.<BR>\n" );
document.write( "From a - b = 0, we have a = b.<BR>\n" );
document.write( "From b - c = 0, we have b = c.<BR>\n" );
document.write( "Substituting b = c into a = b, we get a = c.<BR>\n" );
document.write( "If a = c, then a - c = 0.<BR>\n" );
document.write( "Since a - c = 0, (a, c) ∈ R by the definition of R.  Therefore, R is transitive.\r<BR>\n" );
document.write( "\n" );
document.write( "**Conclusion:**\r<BR>\n" );
document.write( "\n" );
document.write( "Since R is reflexive, symmetric, and transitive, R is an equivalence relation on S = ℝ.  In simpler terms, this relation defines equality:  (a, b) ∈ R if and only if a = b.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );