# MAPLE ASSIGNMENT 1
# 
# SET THEORY
# 
# You can use the following commands to check your understanding of set
# operations. That is, do you agree with Maple's answers?
> S1:={1,2,a,b};
> S2:={a,1,b,2};
> S3:={a,x,y,z,17};
> evalb(S1=S2);
> evalb(S1=S3);
> X:=S1 union S3;
> X1:= S1 intersect S3;
> X2:=S1 minus S3;
> X3:=S3 minus S1;
> X2 intersect X3;
> X4:=X1 union X2 union X3;
> evalb(X4=S1 union S3);
# Define a set S4 which has a nonempty intersection with both S1 and S3
# but is not a subset of X=S1 union S3. Test your answer by (i)
# computing
# S1 intersect S3 intersect S4 and (ii) by using the evalb command to
# show that S4 union X = X is false.
> U:= S1 union S3 union {3,4,5,b,c};
# Calculate the complements of S1, S3, X and X1 with respect to the U
# above, call them cS1, cS3 etc. and use evalb to show that 
# cS1 union cS3 = c(S1 intersect S3) = cX1
# cS1  intersect cS3 = c(S1 union S3) = cX
# 
# The other distributive law for sets is (A union B) intersect C = (A
# intersect C) union (B intersect C). Use S1, S3 and S4 as your sets A,
# B and C in any order. Calculate the LHS and the RHS of the above
# equation and compare the two results to test the law. If you give the
# 2 expressions names you can use evalb to test the law.
# 
# NUMBERS
# 
# The arithmetic operations are represented in Maple by +, -, *,  / and
# ^.
> a:=(3+4)/5;
> b:=(3*4)/5;
> c:=exp(1)^2;
# For precision of calculation, Maple does not convert fractions or
# expressions with exponents unless it is forced to by making one of the
# numbers decimal such as replacing 3 by 3. or by using the evalf
# command.  exp(1) is Maple's way of inputting the number e.
> (3.+4)/5;
> evalf(a);
> evalf(exp(1));
> evalf(Pi);
> 3^(1/3);
> evalf(3^(1/3));
# 
# You can use Maple to check your arithmetical skills.  For example use
# +, * and brackets to write an expression involving 3, 4 and 7 which
# gives
# the answer 49. Does (3/4)+(5/6) = (3+5)/(4+6)?  Check it out. What is
# an alternative expression for calculating (3/4)*(5/6)?
# 
# Maple does symbolic calculations as well as numerical calculations. 
> (x+y)/(w+z);
> (x/w)+(y/z);
> simplify("-"");
# The answer is not zero so the two expressions were not equal. You can
# also use Maple to review the laws of exponents.
> (x^m)+(x^n);
> simplify(");
> (x^m)*(x^n);
> simplify(");
> (x^m)^n;
> simplify(");
# Maple doesn't recognize the second law of exponents amd try the
# following examples to show that Maple can make mistakes.
> (5^(3/2))^(3/4);
> 5^((3/2)*(3/4));
> evalf(");
> 5^(9/8);
> 
>