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Question Number 157887 by Tawa11 last updated on 29/Oct/21

Commented by Tawa11 last updated on 29/Oct/21

Proof please

$$\mathrm{Proof}\:\mathrm{please} \\ $$

Commented by Rasheed.Sindhi last updated on 29/Oct/21

f is not afunction.It′s a relation. Because f(3) is not unique.

$${f}\:{is}\:{not}\:{afunction}.{It}'{s}\:{a}\:{relation}. \\ $$$${Because}\:{f}\left(\mathrm{3}\right)\:{is}\:{not}\:{unique}. \\ $$

Answered by Rasheed.Sindhi last updated on 29/Oct/21

f={(β,a),(3,b),(3,c)} f(β)=a,f(3)=b,f(3)=c g={(a,3),(b,β),(c,β)} g( f(β) )=g(a)=3 g( f(3) )=g(b)=β g( f(3) )=g(c)=β gof={(β,3),(3,β)} g={(a,3),(b,β),(c,β)} g(a)=3,g(b)=β,g(c)=β f={(β,a),(3,b),(3,c)} f( g(a) )=f(3)=b,c f( g(b) )=f(β)=a f( g(c) )=f(β)=a fog={(a,b)(a,c)(b,a),(c,a)}

$${f}=\left\{\left(\beta,{a}\right),\left(\mathrm{3},{b}\right),\left(\mathrm{3},{c}\right)\right\} \\ $$$$\:\:{f}\left(\beta\right)={a},{f}\left(\mathrm{3}\right)={b},{f}\left(\mathrm{3}\right)={c} \\ $$$${g}=\left\{\left({a},\mathrm{3}\right),\left({b},\beta\right),\left({c},\beta\right)\right\} \\ $$$${g}\left(\:{f}\left(\beta\right)\:\right)={g}\left({a}\right)=\mathrm{3} \\ $$$${g}\left(\:{f}\left(\mathrm{3}\right)\:\right)={g}\left({b}\right)=\beta \\ $$$${g}\left(\:{f}\left(\mathrm{3}\right)\:\right)={g}\left({c}\right)=\beta \\ $$$${gof}=\left\{\left(\beta,\mathrm{3}\right),\left(\mathrm{3},\beta\right)\right\} \\ $$$${g}=\left\{\left({a},\mathrm{3}\right),\left({b},\beta\right),\left({c},\beta\right)\right\} \\ $$$$\:\:\:{g}\left({a}\right)=\mathrm{3},{g}\left({b}\right)=\beta,{g}\left({c}\right)=\beta \\ $$$${f}=\left\{\left(\beta,{a}\right),\left(\mathrm{3},{b}\right),\left(\mathrm{3},{c}\right)\right\} \\ $$$${f}\left(\:{g}\left({a}\right)\:\right)={f}\left(\mathrm{3}\right)={b},{c} \\ $$$${f}\left(\:{g}\left({b}\right)\:\right)={f}\left(\beta\right)={a} \\ $$$${f}\left(\:{g}\left({c}\right)\:\right)={f}\left(\beta\right)={a} \\ $$$${fog}=\left\{\left({a},{b}\right)\left({a},{c}\right)\left({b},{a}\right),\left({c},{a}\right)\right\} \\ $$

Commented by Tawa11 last updated on 29/Oct/21

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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