R is an equivalence relation if R is reflexive, symmetric and transitive. The fact that R is symmetric means $\forall x,y\in\mathbb{Z}\,(x,y)\in R\implies (y,x)\in R$. For this definition of $R$ this means that whenever $x=\pm y$, we also have $y=\pm x$. Do you think this is true?mehdi98 said:Suppose that the relation R is defined on the set Z where aRb means a = ±b. Establish whether R is an equivalence relation giving your justifications.
It's easy to see that 3, 8, 13, 18, ... give 3 as a remainder when divided by 5. Therefore, solutions are $3+5k$, $k\in\mathbb{Z}$.mehdi98 said:a) x ≡ 3 (mod 5).
We need to divide both sides by 2. Note that 2 has an inverse modulo 9, i.e., there exists a number $y$ such that $2y$ gives the remainder 1 when divided by 9. Then $2xy\equiv x(2y)\equiv x\equiv 5y\pmod{9}$.mehdi98 said:b) 2x ≡ 5 (mod 9).
A linear congruence is an equation in the form of ax + b ≡ c (mod m), where a, b, and c are constants, x is the variable, and m is the modulus. Solving a linear congruence involves finding the value(s) of x that satisfy the equation.
A relation problem is a question or scenario that involves identifying and analyzing the relationship between two or more variables. This can include determining if the variables are directly or inversely proportional, finding the equation that represents the relationship, and/or solving for unknown values.
To solve a linear congruence, follow these steps:1. Identify the values of a, b, and c in the equation ax + b ≡ c (mod m).2. Determine the value of m, the modulus.3. Use the extended Euclidean algorithm to find the inverse of a modulo m.4. Multiply both sides of the equation by the inverse of a.5. Simplify and solve for x.
A direct relation is one where as one variable increases, the other also increases at a constant rate. In contrast, an inverse relation is one where as one variable increases, the other decreases at a constant rate. In both cases, the relationship can be represented by a linear equation, but the slope will be positive for a direct relation and negative for an inverse relation.
Yes, linear congruence and relation problems can be applied in various real-world situations such as calculating interest rates, determining optimal pricing strategies, and analyzing population growth. These mathematical concepts can also be used in fields such as engineering, economics, and physics.