Proof tasks math student need help
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Can we add congruences? If a≡b mod m and c≡d mod m, is it necessarily true that a+c≡b+d mod m? If so, why? If not, provide an example that illustrates why not. To get started on this question, do some numerical examples.
Can we subtract congruences? If a≡b mod m and c≡d mod m, is it necessarily true that a−c≡b−d mod m? If so, why? If not, provide an example that illustrates why not. To get started on this question, do some numerical examples.
proof-verification proof-explanation
asked Nov 20 at 13:49
S. Totland
22
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up vote
-4
down vote
favorite
Can we add congruences? If a≡b mod m and c≡d mod m, is it necessarily true that a+c≡b+d mod m? If so, why? If not, provide an example that illustrates why not. To get started on this question, do some numerical examples.
Can we subtract congruences? If a≡b mod m and c≡d mod m, is it necessarily true that a−c≡b−d mod m? If so, why? If not, provide an example that illustrates why not. To get started on this question, do some numerical examples.
proof-verification proof-explanation
asked Nov 20 at 13:49
S. Totland
22
1
Welcome to Math.SE, S. Totland! You’ll find that you get a better response here if you give context to your question (where did it come from? Why are you asking about it?) and talk about your thoughts on the problem (what have you tried to solve it? What are the definitions involved?). We try not to answer questions that look like homework problems, or that simply state a question and expect a solution.
– Santana Afton
Nov 20 at 14:03
1
You'll notice that your question already has three downvotes and vote to close. That's almost certainly because it appears to be a direct transcription of a homework problem, and the folks who contribute to this site generally don't like doing people's homework. You can show us what you've tried so far by clicking "edit" beneath your question, and you might get more positive results.
– John Hughes
Nov 20 at 14:04
add a comment |
up vote
-4
down vote
favorite
up vote
-4
down vote
favorite
Can we add congruences? If a≡b mod m and c≡d mod m, is it necessarily true that a+c≡b+d mod m? If so, why? If not, provide an example that illustrates why not. To get started on this question, do some numerical examples.
Can we subtract congruences? If a≡b mod m and c≡d mod m, is it necessarily true that a−c≡b−d mod m? If so, why? If not, provide an example that illustrates why not. To get started on this question, do some numerical examples.
proof-verification proof-explanation
asked Nov 20 at 13:49
S. Totland
22
Can we add congruences? If a≡b mod m and c≡d mod m, is it necessarily true that a+c≡b+d mod m? If so, why? If not, provide an example that illustrates why not. To get started on this question, do some numerical examples.
Can we subtract congruences? If a≡b mod m and c≡d mod m, is it necessarily true that a−c≡b−d mod m? If so, why? If not, provide an example that illustrates why not. To get started on this question, do some numerical examples.
proof-verification proof-explanation
proof-verification proof-explanation
asked Nov 20 at 13:49
S. Totland
22
asked Nov 20 at 13:49
S. Totland
22
edited Nov 20 at 14:02
asked Nov 20 at 13:49
S. Totland
22
asked Nov 20 at 13:49
S. Totland
22
asked Nov 20 at 13:49
S. Totland
22
22
1
Welcome to Math.SE, S. Totland! You’ll find that you get a better response here if you give context to your question (where did it come from? Why are you asking about it?) and talk about your thoughts on the problem (what have you tried to solve it? What are the definitions involved?). We try not to answer questions that look like homework problems, or that simply state a question and expect a solution.
– Santana Afton
Nov 20 at 14:03
1
You'll notice that your question already has three downvotes and vote to close. That's almost certainly because it appears to be a direct transcription of a homework problem, and the folks who contribute to this site generally don't like doing people's homework. You can show us what you've tried so far by clicking "edit" beneath your question, and you might get more positive results.
– John Hughes
Nov 20 at 14:04
add a comment |
1
Welcome to Math.SE, S. Totland! You’ll find that you get a better response here if you give context to your question (where did it come from? Why are you asking about it?) and talk about your thoughts on the problem (what have you tried to solve it? What are the definitions involved?). We try not to answer questions that look like homework problems, or that simply state a question and expect a solution.
– Santana Afton
Nov 20 at 14:03
1
You'll notice that your question already has three downvotes and vote to close. That's almost certainly because it appears to be a direct transcription of a homework problem, and the folks who contribute to this site generally don't like doing people's homework. You can show us what you've tried so far by clicking "edit" beneath your question, and you might get more positive results.
– John Hughes
Nov 20 at 14:04
1
1
Welcome to Math.SE, S. Totland! You’ll find that you get a better response here if you give context to your question (where did it come from? Why are you asking about it?) and talk about your thoughts on the problem (what have you tried to solve it? What are the definitions involved?). We try not to answer questions that look like homework problems, or that simply state a question and expect a solution.
– Santana Afton
Nov 20 at 14:03
Welcome to Math.SE, S. Totland! You’ll find that you get a better response here if you give context to your question (where did it come from? Why are you asking about it?) and talk about your thoughts on the problem (what have you tried to solve it? What are the definitions involved?). We try not to answer questions that look like homework problems, or that simply state a question and expect a solution.
– Santana Afton
Nov 20 at 14:03
1
1
You'll notice that your question already has three downvotes and vote to close. That's almost certainly because it appears to be a direct transcription of a homework problem, and the folks who contribute to this site generally don't like doing people's homework. You can show us what you've tried so far by clicking "edit" beneath your question, and you might get more positive results.
– John Hughes
Nov 20 at 14:04
You'll notice that your question already has three downvotes and vote to close. That's almost certainly because it appears to be a direct transcription of a homework problem, and the folks who contribute to this site generally don't like doing people's homework. You can show us what you've tried so far by clicking "edit" beneath your question, and you might get more positive results.
– John Hughes
Nov 20 at 14:04
add a comment |
2 Answers
2
active
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up vote
3
down vote
Usually we talk about adding congruence classes. For example, if $a=b$ mod(m), we usually say the congruence classes of a and b are equal, that is $[a]=[b]$, and we can add/subtract/multiply two congruence classes.
I'll show the first question and you can do the second one.
By definition, $a=b$ mod(m) $iff$ $a-b=0$ mod(m) $iff$ $a-b=tm$ for some $tinmathbb{Z}$.
Using this fact, we can see that $$a-b+c-d=tm+sm=(t+s)m$$ for some $t,sinmathbb{Z}$. If we add $b+d$ to both sides, we obtain $$a+c=b+d+(t+s)m.$$ Taking mod(m) of both sides, we obtain our result.
answered Nov 20 at 14:04
Blake Jackson
765
1
+1 This is correct and well written. That said, I'd rather you waited to answer until the OP showed some effort of his or her own - note the comments on the questions.
– Ethan Bolker
Nov 20 at 14:45
@EthanBolker Noted. Thank you for the input!
– Blake Jackson
Nov 20 at 18:46
add a comment |
up vote
1
down vote
Consider: $$a≡b mod(m) hspace{0.5cm} and hspace{0.5cm} c≡d mod(m)$$
Then, $$m mid (a-b) hspace{0.5cm} and hspace{0.5cm} m mid (c-d)$$
So, $$a-b=km hspace{0.5cm} and hspace{0.5cm} c-d=k'm$$
Adding, $$(a+c)-(b+d)=(k+k')m$$
Hence, $$ m mid (a+c)-(b+d)$$
We get: $$(a+c)≡(b+d) mod m$$
Similarly, proceed for: $$(a-c)≡(b-d) mod m$$
You wanted numerical examples:
Take $$7≡5 mod(2) hspace{0.5cm} and hspace{0.5cm} 10≡4 mod(2)$$
$$7+10≡5+4 mod(2)$$ or $$10≡4 mod(2)$$ $$17≡9 mod(2)$$
Similarly,
$$7-10≡5-4 mod(2)$$ or $$-3≡1 mod(2)$$
answered Nov 20 at 14:15
idea
1
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Usually we talk about adding congruence classes. For example, if $a=b$ mod(m), we usually say the congruence classes of a and b are equal, that is $[a]=[b]$, and we can add/subtract/multiply two congruence classes.
I'll show the first question and you can do the second one.
By definition, $a=b$ mod(m) $iff$ $a-b=0$ mod(m) $iff$ $a-b=tm$ for some $tinmathbb{Z}$.
Using this fact, we can see that $$a-b+c-d=tm+sm=(t+s)m$$ for some $t,sinmathbb{Z}$. If we add $b+d$ to both sides, we obtain $$a+c=b+d+(t+s)m.$$ Taking mod(m) of both sides, we obtain our result.
answered Nov 20 at 14:04
Blake Jackson
765
1
+1 This is correct and well written. That said, I'd rather you waited to answer until the OP showed some effort of his or her own - note the comments on the questions.
– Ethan Bolker
Nov 20 at 14:45
@EthanBolker Noted. Thank you for the input!
– Blake Jackson
Nov 20 at 18:46
add a comment |
up vote
3
down vote
Usually we talk about adding congruence classes. For example, if $a=b$ mod(m), we usually say the congruence classes of a and b are equal, that is $[a]=[b]$, and we can add/subtract/multiply two congruence classes.
I'll show the first question and you can do the second one.
By definition, $a=b$ mod(m) $iff$ $a-b=0$ mod(m) $iff$ $a-b=tm$ for some $tinmathbb{Z}$.
Using this fact, we can see that $$a-b+c-d=tm+sm=(t+s)m$$ for some $t,sinmathbb{Z}$. If we add $b+d$ to both sides, we obtain $$a+c=b+d+(t+s)m.$$ Taking mod(m) of both sides, we obtain our result.
answered Nov 20 at 14:04
Blake Jackson
765
1
+1 This is correct and well written. That said, I'd rather you waited to answer until the OP showed some effort of his or her own - note the comments on the questions.
– Ethan Bolker
Nov 20 at 14:45
@EthanBolker Noted. Thank you for the input!
– Blake Jackson
Nov 20 at 18:46
add a comment |
up vote
3
down vote
up vote
3
down vote
Usually we talk about adding congruence classes. For example, if $a=b$ mod(m), we usually say the congruence classes of a and b are equal, that is $[a]=[b]$, and we can add/subtract/multiply two congruence classes.
I'll show the first question and you can do the second one.
By definition, $a=b$ mod(m) $iff$ $a-b=0$ mod(m) $iff$ $a-b=tm$ for some $tinmathbb{Z}$.
Using this fact, we can see that $$a-b+c-d=tm+sm=(t+s)m$$ for some $t,sinmathbb{Z}$. If we add $b+d$ to both sides, we obtain $$a+c=b+d+(t+s)m.$$ Taking mod(m) of both sides, we obtain our result.
answered Nov 20 at 14:04
Blake Jackson
765
Usually we talk about adding congruence classes. For example, if $a=b$ mod(m), we usually say the congruence classes of a and b are equal, that is $[a]=[b]$, and we can add/subtract/multiply two congruence classes.
I'll show the first question and you can do the second one.
By definition, $a=b$ mod(m) $iff$ $a-b=0$ mod(m) $iff$ $a-b=tm$ for some $tinmathbb{Z}$.
Using this fact, we can see that $$a-b+c-d=tm+sm=(t+s)m$$ for some $t,sinmathbb{Z}$. If we add $b+d$ to both sides, we obtain $$a+c=b+d+(t+s)m.$$ Taking mod(m) of both sides, we obtain our result.
answered Nov 20 at 14:04
Blake Jackson
765
answered Nov 20 at 14:04
Blake Jackson
765
answered Nov 20 at 14:04
Blake Jackson
765
answered Nov 20 at 14:04
Blake Jackson
765
765
1
+1 This is correct and well written. That said, I'd rather you waited to answer until the OP showed some effort of his or her own - note the comments on the questions.
– Ethan Bolker
Nov 20 at 14:45
@EthanBolker Noted. Thank you for the input!
– Blake Jackson
Nov 20 at 18:46
add a comment |
1
+1 This is correct and well written. That said, I'd rather you waited to answer until the OP showed some effort of his or her own - note the comments on the questions.
– Ethan Bolker
Nov 20 at 14:45
@EthanBolker Noted. Thank you for the input!
– Blake Jackson
Nov 20 at 18:46
1
1
+1 This is correct and well written. That said, I'd rather you waited to answer until the OP showed some effort of his or her own - note the comments on the questions.
– Ethan Bolker
Nov 20 at 14:45
+1 This is correct and well written. That said, I'd rather you waited to answer until the OP showed some effort of his or her own - note the comments on the questions.
– Ethan Bolker
Nov 20 at 14:45
@EthanBolker Noted. Thank you for the input!
– Blake Jackson
Nov 20 at 18:46
@EthanBolker Noted. Thank you for the input!
– Blake Jackson
Nov 20 at 18:46
add a comment |
up vote
1
down vote
Consider: $$a≡b mod(m) hspace{0.5cm} and hspace{0.5cm} c≡d mod(m)$$
Then, $$m mid (a-b) hspace{0.5cm} and hspace{0.5cm} m mid (c-d)$$
So, $$a-b=km hspace{0.5cm} and hspace{0.5cm} c-d=k'm$$
Adding, $$(a+c)-(b+d)=(k+k')m$$
Hence, $$ m mid (a+c)-(b+d)$$
We get: $$(a+c)≡(b+d) mod m$$
Similarly, proceed for: $$(a-c)≡(b-d) mod m$$
You wanted numerical examples:
Take $$7≡5 mod(2) hspace{0.5cm} and hspace{0.5cm} 10≡4 mod(2)$$
$$7+10≡5+4 mod(2)$$ or $$10≡4 mod(2)$$ $$17≡9 mod(2)$$
Similarly,
$$7-10≡5-4 mod(2)$$ or $$-3≡1 mod(2)$$
answered Nov 20 at 14:15
idea
1
add a comment |
up vote
1
down vote
Consider: $$a≡b mod(m) hspace{0.5cm} and hspace{0.5cm} c≡d mod(m)$$
Then, $$m mid (a-b) hspace{0.5cm} and hspace{0.5cm} m mid (c-d)$$
So, $$a-b=km hspace{0.5cm} and hspace{0.5cm} c-d=k'm$$
Adding, $$(a+c)-(b+d)=(k+k')m$$
Hence, $$ m mid (a+c)-(b+d)$$
We get: $$(a+c)≡(b+d) mod m$$
Similarly, proceed for: $$(a-c)≡(b-d) mod m$$
You wanted numerical examples:
Take $$7≡5 mod(2) hspace{0.5cm} and hspace{0.5cm} 10≡4 mod(2)$$
$$7+10≡5+4 mod(2)$$ or $$10≡4 mod(2)$$ $$17≡9 mod(2)$$
Similarly,
$$7-10≡5-4 mod(2)$$ or $$-3≡1 mod(2)$$
answered Nov 20 at 14:15
idea
1
add a comment |
up vote
1
down vote
up vote
1
down vote
Consider: $$a≡b mod(m) hspace{0.5cm} and hspace{0.5cm} c≡d mod(m)$$
Then, $$m mid (a-b) hspace{0.5cm} and hspace{0.5cm} m mid (c-d)$$
So, $$a-b=km hspace{0.5cm} and hspace{0.5cm} c-d=k'm$$
Adding, $$(a+c)-(b+d)=(k+k')m$$
Hence, $$ m mid (a+c)-(b+d)$$
We get: $$(a+c)≡(b+d) mod m$$
Similarly, proceed for: $$(a-c)≡(b-d) mod m$$
You wanted numerical examples:
Take $$7≡5 mod(2) hspace{0.5cm} and hspace{0.5cm} 10≡4 mod(2)$$
$$7+10≡5+4 mod(2)$$ or $$10≡4 mod(2)$$ $$17≡9 mod(2)$$
Similarly,
$$7-10≡5-4 mod(2)$$ or $$-3≡1 mod(2)$$
answered Nov 20 at 14:15
idea
1
Consider: $$a≡b mod(m) hspace{0.5cm} and hspace{0.5cm} c≡d mod(m)$$
Then, $$m mid (a-b) hspace{0.5cm} and hspace{0.5cm} m mid (c-d)$$
So, $$a-b=km hspace{0.5cm} and hspace{0.5cm} c-d=k'm$$
Adding, $$(a+c)-(b+d)=(k+k')m$$
Hence, $$ m mid (a+c)-(b+d)$$
We get: $$(a+c)≡(b+d) mod m$$
Similarly, proceed for: $$(a-c)≡(b-d) mod m$$
You wanted numerical examples:
Take $$7≡5 mod(2) hspace{0.5cm} and hspace{0.5cm} 10≡4 mod(2)$$
$$7+10≡5+4 mod(2)$$ or $$10≡4 mod(2)$$ $$17≡9 mod(2)$$
Similarly,
$$7-10≡5-4 mod(2)$$ or $$-3≡1 mod(2)$$
answered Nov 20 at 14:15
idea
1
edited Nov 20 at 14:24
answered Nov 20 at 14:15
idea
1
answered Nov 20 at 14:15
idea
1
answered Nov 20 at 14:15
idea
1
1
add a comment |
add a comment |
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1
Welcome to Math.SE, S. Totland! You’ll find that you get a better response here if you give context to your question (where did it come from? Why are you asking about it?) and talk about your thoughts on the problem (what have you tried to solve it? What are the definitions involved?). We try not to answer questions that look like homework problems, or that simply state a question and expect a solution.
– Santana Afton
Nov 20 at 14:03
1
You'll notice that your question already has three downvotes and vote to close. That's almost certainly because it appears to be a direct transcription of a homework problem, and the folks who contribute to this site generally don't like doing people's homework. You can show us what you've tried so far by clicking "edit" beneath your question, and you might get more positive results.
– John Hughes
Nov 20 at 14:04