06-240/Homework Assignment 1 - Revision history

Drorbn: Reverted edit of 199.164.125.138, changed back to last version by Drorbn

2007-06-14T15:11:24Z

Reverted edit of 199.164.125.138, changed back to last version by Drorbn

← Older revision		Revision as of 11:11, 14 June 2007
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	# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.		# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
	# Write the following complex numbers in the form <math>a ib</math>, with <math>a,b\in{\mathbb R}</math>:		# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:
	## <math>\frac{1}{2i} \frac{-2i}{5-i}</math>.		## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.
	## <math>(1 i)^5</math>.		## <math>(1+i)^5</math>.
	#		#
	## Prove that the set <math>F_1=\{a b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.		## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
	## Is the set <math>F_2=\{a b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?		## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?
	# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1 1=0</math>. Prove that <math>b=a^{-1}=a^2=a 1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)		# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)

	This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

199.164.125.138 at 05:19, 14 June 2007

2007-06-14T05:19:03Z

← Older revision		Revision as of 01:19, 14 June 2007
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	# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.		# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
	# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:		# Write the following complex numbers in the form <math>a ib</math>, with <math>a,b\in{\mathbb R}</math>:
	## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.		## <math>\frac{1}{2i} \frac{-2i}{5-i}</math>.
	## <math>(1+i)^5</math>.		## <math>(1 i)^5</math>.
	#		#
	## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.		## Prove that the set <math>F_1=\{a b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
	## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?		## Is the set <math>F_2=\{a b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?
	# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)		# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1 1=0</math>. Prove that <math>b=a^{-1}=a^2=a 1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)

	This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn: Reverted edit of 59.94.238.228, changed back to last version by Drorbn

2007-05-28T14:50:21Z

Reverted edit of 59.94.238.228, changed back to last version by Drorbn

← Older revision		Revision as of 10:50, 28 May 2007
Line 4:		Line 4:

	# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.		# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
	# Write the following complex numbers in the form <math>a ib</math>, with <math>a,b\in{\mathbb R}</math>:		# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:
	## <math>\frac{1}{2i} \frac{-2i}{5-i}</math>.		## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.
	## <math>(1 i)^5</math>.		## <math>(1+i)^5</math>.
	#		#
	## Prove that the set <math>F_1=\{a b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.		## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
	## Is the set <math>F_2=\{a b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?		## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?
	# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1 1=0</math>. Prove that <math>b=a^{-1}=a^2=a 1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)		# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)

	This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

59.94.238.228 at 06:38, 28 May 2007

2007-05-28T06:38:32Z

← Older revision		Revision as of 02:38, 28 May 2007
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	# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.		# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
	# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:		# Write the following complex numbers in the form <math>a ib</math>, with <math>a,b\in{\mathbb R}</math>:
	## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.		## <math>\frac{1}{2i} \frac{-2i}{5-i}</math>.
	## <math>(1+i)^5</math>.		## <math>(1 i)^5</math>.
	#		#
	## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.		## Prove that the set <math>F_1=\{a b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
	## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?		## Is the set <math>F_2=\{a b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?
	# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)		# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1 1=0</math>. Prove that <math>b=a^{-1}=a^2=a 1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)

	This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn at 22:09, 13 September 2006

2006-09-13T22:09:22Z

← Older revision		Revision as of 18:09, 13 September 2006
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	## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.		## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
	## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?		## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?
	# Let <math>F=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)		# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)

	This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn at 22:03, 13 September 2006

2006-09-13T22:03:03Z

← Older revision		Revision as of 18:03, 13 September 2006
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	# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.		# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
⚫	# Let <math>F=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)
	# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:		# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:
	## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.		## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.
	## <math>(1+i)^5</math>.		## <math>(1+i)^5</math>.
			#
			## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
			## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?
		⚫	# Let <math>F=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)

	This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work; it will be ~~your~~ ~~problem,~~ ~~not~~ ~~theirs~~.		This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn at 21:54, 13 September 2006

2006-09-13T21:54:36Z

← Older revision		Revision as of 17:54, 13 September 2006
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	Read appendices A through D in our textbook (with higher attention to C and D), and solve the following problems:		Read appendices A through D in our textbook (with higher attention to C and D), and solve the following problems:

			# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
			# Let <math>F=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)
			# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:
			## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.
			## <math>(1+i)^5</math>.

			This assignment is due at the tutorials on Thursday September 21. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work; it will be your problem, not theirs.

Drorbn at 21:40, 13 September 2006

2006-09-13T21:40:18Z

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Read appendices A through D in our textbook (with higher attention to C and D), and solve the following problems: