Let R be the relation on Z×Z ,that is elements of this relat
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Let R be the relation on Z×Z ,that is elements of this relation are
pairs of pairs of integers,such that ((a,b),(c,d)) ∈ R if and only if
a+d = b+c.Is R an equivalence relation?Explain.
pairs of pairs of integers,such that ((a,b),(c,d)) ∈ R if and only if
a+d = b+c.Is R an equivalence relation?Explain.
先证递移性 {(a,b),(c,d)}∈ R {(c,d),(e,f)}∈ R=> a+d=b+c;c+f=d+e
=>a+d+c+f=b+c+d+e => a+f=b+e =>{(a,b),(e,f)}∈ R ...ok
再证对称性 {(a,b),(c,d)}∈ R =>a+d=b+c =>c+b=d+a =>{(c,d),(a,b)}∈ R
.OK
最后证反身行性{(a,b),(a,b)}∈ R => a+b=a+b 显然
=>R是等价关系
=>a+d+c+f=b+c+d+e => a+f=b+e =>{(a,b),(e,f)}∈ R ...ok
再证对称性 {(a,b),(c,d)}∈ R =>a+d=b+c =>c+b=d+a =>{(c,d),(a,b)}∈ R
.OK
最后证反身行性{(a,b),(a,b)}∈ R => a+b=a+b 显然
=>R是等价关系
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