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Thread: Q: Let S = {x, y, z } and R is a relation defined on S such that?

  1. #1
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    Q: Let S = {x, y, z } and R is a relation defined on S such that?

    Q: Let S = {x, y, z } and R is a relation defined on S such that
    R={(y,y),(x,z),(z,x),(x,x),(z,z),(x,y),(y,x)}
    Show that R is reflexive and symmetric as well.
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    Re: Q: Let S = {x, y, z } and R is a relation defined on S such that?

    Since R is finite, you can easily check this by brute force. R is reflexive because (x,x), (y,y), and (z,z) are in R.
    For symmetry: Since (x,z) is in R, we also need (z,x) in R. Is it? Yes it is. Similarly, since (x,y) is in R, we need (y,x) in R, and indeed, it is.
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    Re: Q: Let S = {x, y, z } and R is a relation defined on S such that?

    Quote Originally Posted by sMilips View Post
    Q: Let S = {x, y, z } and R is a relation defined on S such that
    R={(y,y),(x,z),(z,x),(x,x),(z,z),(x,y),(y,x)}
    Show that R is reflexive and symmetric as well.
    The diagonal of a set $\Delta_S=\{(x,x): x\in S\}$
    A relation $\mathcal{R}$ is reflexive on $\mathit{S}$ iff $\Delta_S\subseteq\mathcal{R}$
    A relation $\mathcal{R}$ is symmetric on $\mathit{S}$ iff $\mathcal{R}=\mathcal{R}^{-1}$
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