Math Theorems
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27 terms
Terms | Definitions |
|---|---|
 Addition Property of Equality (Thm 1) | For all a, b, c, in R if a=b, then a+c=b+c |
| Cancellation Property of Addition (Thm 2) | For all a, b, c, in R if a+c=b+c, then a=b |
| Opposites of Sums (Thm 3) | For all a, b, in R, -(a+b)=(-a)+(-b) |
| Opposites of Opposites (Thm 4) | For all a in R, -(-a)=a |
| The Multiplicative Property of Equality (Thm 5) | For all a, b, c, in R if a=b then ac=bc |
| Cancellation Property of Multiplication (Thm 6) | For all a, b, c, in R if ac=bc and c is not equal to 0, then a=b |
| Reciprocals of Products (Thm 7) | For all a, b, in R where a,b are not equal to 0, 1/ab=(1/a) (1/b) |
| Reciprocals of Reciprocals (Thm 8) | For all a in R where a is not equal to 0, a=1/1/a |
| Multiplicative Property of Zero (Thm 9) | For all x in R x*0=0 |
| Converse of the Multiplicative Property of Zero (Thm 10) | For all a,b in R if ab=0, then a=0 or b=0 |
| Multiplicative Property of (-1) (Thm 11) | For all a in R (-1)a=-a |
| Multiplying by Opposites (Thm 12) | For all a,b in R (-a)b=a(-b)=-(ab) |
| Subtracting the Opposite (Thm 13) | For all a,b in R a-(-b)=a+b |
| Non-Commutative Property of Subtraction (Thm 14) | For all a,b in R a-b= -(b-a) |
| Non-Associative Property of Subtraction (Thm 15) | For all a,b,c in R a-(b-c)=(a-b)+c |
| Dividing by the Reciprocal (Thm 16) | For all a,b in R where b is not equal to 0 a/1/b=a*b |
| Non-Commutative Law for Division (Thm 17) | For all a,b in R where a,b are not equal to 0, 1/a/b=b/a |
| Non-Associative Law for Division (Thm 18) | For all a,b,c in R where b,c are not equal to 0, a/(b/c)=ac/b |
| Multiplication of "Fractions" (Thm 19) | For all a,b,c,d where b,d are not equal to 0, a/b*c/d=ac/bd |
| Division by "Fractions" (Thm 20) | For all a,b,c,d in R where b,c,d are not equal to 0, a/b/c/d=ad/bc |
| The Transitivity of Inequality (Thm 21) | If a<b and b<c, then a<c |
| The Addition Property of Inequality (Thm 22) | For any real numbers a,b,c, if a<b, then a+c<b+c |
| The Multiplication Property of Inequality (Thm 23) | A) If a<b and c>0 then ac<bc B) If a<b and c<0 then ac>bc |
| The Inequality of the Opposite (Thm 24) | If a>0, then -a<0 |
| The Inequality of the reciprocal (Thm 25) | If a>1 then 0<1/a<1 |
| The Square is Never Negative (Thm 26) | For all real numbers a, a²≥0 |
| Inequalities Added (Thm 27) | If a<b and c<d, then a+c<b+d |
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