1. **Problem Statement:** Given the relation B on the set $\{1, 2, 3, 4, 5\}$ defined by the pairs $\{(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (5, 1), (5, 3), (5, 5)\}$, represent it as a matrix. 2. **Matrix Representation:** The matrix $M$ of a relation on a set with elements $\{1, 2, 3, 4, 5\}$ is a $5 \times 5$ matrix where the entry $M_{ij} = 1$ if $(i, j)$ is in the relation, and $0$ otherwise. 3. **Constructing the matrix:** - For $i=1$: pairs are $(1,1), (1,3), (1,5)$ so row 1 is $[1, 0, 1, 0, 1]$ - For $i=2$: pairs are $(2,2), (2,4)$ so row 2 is $[0, 1, 0, 1, 0]$ - For $i=3$: pairs are $(3,1), (3,3), (3,5)$ so row 3 is $[1, 0, 1, 0, 1]$ - For $i=4$: pairs are $(4,2), (4,4)$ so row 4 is $[0, 1, 0, 1, 0]$ - For $i=5$: pairs are $(5,1), (5,3), (5,5)$ so row 5 is $[1, 0, 1, 0, 1]$ 4. **Final matrix:** $$ M = \begin{bmatrix} 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \end{bmatrix} $$ This matrix fully represents the relation B. **Answer:** The matrix representation of relation B is as above.