What is the solution set for the open sentence with the given replacement set? 2t-t=0,{1,2,3,4} a. null set b. 2 c. 3 d. 4

Answer is choice A, the null set

This is another way of saying "the empty set" which is the set of nothing inside it. It's a fancy mathematical way of saying "no solutions" in terms of the given restrictions placed upon us.



Let's check every value in the replacement set. All we do is go through each number one at a time

Checking t = 1
2t - t = 0
2*1 - 1 = 0 ... replace every t with 1
2 - 1 = 0
1 = 0
The last equation is false so we can cross t=1 off the list

Checking t = 2
2t - t = 0
2*2 - 2 = 0 ... replace every t with 2
4 - 2 = 0
2 = 0
The last equation is false so we can cross t=2 off the list

Checking t = 3
2t - t = 0
2*3 - 3 = 0 ... replace every t with 3
6 - 3 = 0
3 = 0
The last equation is false so we can cross t=3 off the list

Checking t = 4
2t - t = 0
2*4 - 4 = 0 ... replace every t with 4
8 - 4 = 0
4 = 0
The last equation is false so we can cross t=4 off the list

We've crossed everything off the list. So there are no solutions indicating the solution set is the null set or empty set.

Side Note: if t = 0 were in the replacement set, then we'd have a solution. However that isn't the case here.