Find the pairs of equal sets, if any, give reasons:\ A = \{0\} B = \{x : x > 15 and x<5C={x:x−5=0}D={x: x2=25} E = \{x : x is an integral positive root of the equation x2 - 2x - 15 = 0\}
Get the complete, step-by-step math solution for: "Find the pairs of equal sets, if any, give reasons:\ A = \{0\} B = \{x : x > 15 and x < 5} C = \{x : x - 5 = 0 \} D = \{x: x² = 25} E = \{x : x is an ...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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 x2=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 x2−2x−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.