Relations and Functions

If $\left(\frac{x}{3} + 1, y - \frac{2}{3}\right) = \left(\frac{5}{3}, \frac{1}{3}\right)$, find the values of $x$ and $y$.

If the set $A$ has 3 elements and the set $B = \{3, 4, 5\}$, then find the number of elements in $(A \times B)$.

If $G = \{7, 8\}$ and $H = \{5, 4, 2\}$, find $G \times H$ and $H \times G$.

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.

If $A = \{-1, 1\}$, find $A \times A \times A$.

If $A \times B = \{(a, x), (a, y), (b, x), (b, y)\}$. Find $A$ and $B$.

Let $A = \{1, 2\}$, $B = \{1, 2, 3, 4\}$, $C = \{5, 6\}$ and $D = \{5, 6, 7, 8\}$. Verify that

Let $A = \{1, 2\}$ and $B = \{3, 4\}$. Write $A \times B$. How many subsets will $A \times B$ have? List them.

Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 2$. If $(x, 1), (y, 2), (z, 1)$ are in $A \times B$, find $A$ and $B$, where $x, y$ and $z$ are distinct elements.

The Cartesian product $A \times A$ has 9 elements among which are found $(-1, 0)$ and $(0, 1)$. Find the set $A$ and the remaining elements of $A \times A$.

Let $A = \{1, 2, 3, \ldots, 14\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y): 3x - y = 0, \text{where } x, y \in A\}$. Write down its domain, codomain and range.

Define a relation $R$ on the set $N$ of natural numbers by $R = \{(x, y) : y = x + 5, x \text{ is a natural number less than } 4; x, y \in N\}$. Depict this relationship using roster form. Write down the domain and the range.

$A = \{1, 2, 3, 5\}$ and $B = \{4, 6, 9\}$. Define a relation $R$ from $A$ to $B$ by $R = \{(x, y) : \text{the difference between } x \text{ and } y \text{ is odd; } x \in A, y \in B\}$. Write $R$ in roster form.

Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b) : a, b \in A, b \text{ is exactly divisible by } a\}$.

Determine the domain and range of the relation $R$ defined by $R = \{(x, x + 5) : x \in \{0, 1, 2, 3, 4, 5\}\}$.

Write the relation $R = \{(x, x^3) : x \text{ is a prime number less than } 10\}$ in roster form.

Let $A = \{x, y, z\}$ and $B = \{1, 2\}$. Find the number of relations from $A$ to $B$.

Let $R$ be the relation on $Z$ defined by $R = \{(a, b) : a, b \in Z, a - b \text{ is an integer}\}$. Find the domain and range of $R$.

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

Find the domain and range of the following real functions: