Proof Tutorial Class, Week 23

Question One §

Question §

Draw the mapping $g:\mathbb{R}\to \mathbb{R}$ on the $(x,y)$ plane defined by the rule following rule - is it injective, surjective?

$$g(x)=\text{largest integer}\le x$$

Solution §

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Question Two §

Question §

For the following mapping, what is the image of $g$? What is the full inverse image $g^{-1}(2)$?

$$g(x)=\text{largest integer}\le x$$

Solution §

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Question Three §

Question §

Given any mapping $f:A\to B$, prove that the relation $\sim$ on $A$ defined by $a_1\sim a_2$ if $f(a_1)=f(a_2)$ is an equivalence relation. What are the equivalence classes?

Solution §

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