1. The problem asks to determine which of the given relations define functions from set E to set F. 2. Recall that a relation $R$ from $E$ to $F$ defines a function if and only if every element in $E$ is related to exactly one element in $F$. 3. For relation $\Gamma_{R1} = \{(2,1), (6,3)\}$ with $E = \{0,1,2,5,6\}$ and $F = \{1,2,3\}$: - Elements 0, 1, and 5 in $E$ have no image in $F$. - Elements 2 and 6 have exactly one image. - Since not all elements of $E$ have an image, $\Gamma_{R1}$ does not define a function. 4. For relation $\Gamma_{R2} = \{(1,3), (1,5), (2,5)\}$ with $E = \{1,2,3,4\}$ and $F = \{3,5,6\}$: - Element 1 in $E$ is related to two elements in $F$ (3 and 5). - This violates the definition of a function. - Therefore, $\Gamma_{R2}$ does not define a function. 5. For relation $\Gamma_{R3} = \{(1,c), (2,b), (3,a), (4,b)\}$ with $E = \{1,2,3,4\}$ and $F = \{a,b,c\}$: - Each element in $E$ is related to exactly one element in $F$. - Therefore, $\Gamma_{R3}$ defines a function from $E$ to $F$. Final answer: - $\Gamma_{R1}$ is not a function. - $\Gamma_{R2}$ is not a function. - $\Gamma_{R3}$ is a function.