Prove or give a counterexample to each of the following statements. \\begin{enum

Question

Prove or give a counterexample to each of the following statements. egin{enumerate} item For each non-negative number s, there exists a non-negative number t such that $s geq t$. item There exists a non-negative number t such that for all non-negative numbers s, the inequality$ s geq t $holds. item For each non-negative number t, there exists a non-negative number s such that$ s geq t $ item There exists a non-negative number s such that for all non-negative numbers t, the inequality $s geq t $holds.

Explanation / Answer

(The question is difficult to read!)

For each non-negative number s, there exists a non-negative number t such that s >= t.

Proof: Consider the number s/10.

Since s >= 0, s/10 >= 0

s >= s/10

Thus the given statement is true as there exists t = s/10 which is non-negative and s >= t.

There exists a non-negative number t such that for all non-negative numbers s, the inequality s >= t holds.

Disproof by counterexample: Consider s = t/10.

Since t > = 0, t/10 >= 0.

Also t >= t/10.

Thus there is atleast one number s = t/10 which is non-negative and the inequality s >= t does not hold. The given statement is false.

For each non-negative number t, there exists a non-negative number s such that s >= t

Proof: Consider the number t*10.

Since t >= 0, t*10 >= 0

t*10 >= t

Thus the given statement is true as there exists s = 10t which is non-negative and s >= t.

There exists a non-negative number s such that for all non-negative numbers t, the inequality s >= t holds.

Disproof by counterexample: Consider t = s*10.

Since s > = 0, s*10 >= 0.

Also s*10 >= s.

Thus there is atleast one number t = 10s which is non-negative and the inequality s >= t does not hold. The given statement is false.

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