Find the pairs of equal sets, if any, give reasons:\ A = \{0\} B = \{x : x > 15 $\text{ and } x < 5$} C = \{x : x - 5 = 0 \} D = \{x: $x^2 = 25$} E = \{x : x $\text{ is an integral positive root of the equation } x^2 - 2x - 15 = 0$}

Step 1: Determine elements of Set A

Set A is explicitly defined as containing only one element, which is 0. No further calculation is needed for this set.

Step 2: Determine elements of Set B

Set B is defined by the condition that $x$ must be greater than 15 AND less than 5 simultaneously. There is no number that can satisfy both conditions at the same time. Therefore, Set B is an empty set.

Step 3: Determine elements of Set C

Set C is defined by the equation $x - 5 = 0$. Solving this linear equation for $x$ gives $x = 5$. Thus, Set C contains only the element 5.

Step 4: Determine elements of Set D

Set D is defined by the equation $x^2 = 25$. Taking the square root of both sides yields two possible values for $x$: positive 5 and negative 5. Therefore, Set D contains both 5 and -5.

Step 5: Determine elements of Set E

Set E requires finding the integral positive root of the quadratic equation $x^2 - 2x - 15 = 0$. Factoring the quadratic equation gives $(x-5)(x+3) = 0$, which means $x=5$ or $x=-3$. Since we are looking for an integral positive root, $x=5$ is the only valid solution. Thus, Set E contains only the element 5.

Step 6: Identify equal sets

By comparing the elements of all the sets, we find that Set C contains only 5, and Set E also contains only 5. Therefore, Set C and Set E are equal sets.

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