Question

The graph of a relation is
(i) Symmetric with respect to the $x$-axis provided that whenever (a, b) is a point on the graph, so is $(a,-b)$
(ii) Symmetric with respect to the $y$-axis provided that whenever $(a,b)$ is a point on the graph, so is $(-a,b)$
(iii) Symmetric with respect to the origin provided that whenever (a, b) is a point on the graph, so is $(-a,-b)$
(iv) Symmetric with respect to the line $y=x$, provided that whenever $(a,b)$ is a point on the graph, so is $(b,a)$
Suppose R is a relation whose graph is symmetric to both the x-axis and y-axis, and that the point (1, 2) is on the graph of R. Which one of the following points is NOT necessarily on the graph of R?

• A. (-1, 2)
• B. (1, - 2)
• C. (-1, -2)
• D. (2, 1)

Answer 1

Answer: (D)

Suppose $R$ is just a rectangle whose 4 vertices are $(1,2),(1,-2),(-1,2)$ and $(-1,-2)$. The $x$-axis and $y$-axis symmetries in the problem are satisfied, but the point $(2,1)$ is not contained in $R$.]