I have 2 questions for homework that I think I know the answer but im really not sure. We are given that f:A→B is a function and C1 and C2 are subsets of A
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Now I need to prove if this 2 statement are true or false:
1. If f(C1)⊂f(C2) then C1⊂C2 .
2. If f is an injective function and f(C1)⊂f(C2), then C1⊂C2 .
I do (1), you then get the point and you do (2):
Take the function $\;f:\{1,2,3,4\}\to\{a,b,c\}\;$ defined as $\;f(1)=f(2)=a\,,\,f(3)=b\,,\,f(4)=c\;$, and now take the subsets
$$C_1=\{1,4\}\;,\;\;C_2=\{2,4\}\implies f(C_1)=\{a,c\}\subset\{a,c\}=F(C_2)$$
yet $\;C_1\rlap{\;\,/}\subset C_2\;$ (nor the other way around, by the way)