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Question
If A = {3, 5, 7, 9, 11, 12}, determine the truth value of the following.
∀ x ∈ A, 2x + 9 > 14
Solution
This means for every x in A, the inequality 2x + 9 > 14 must hold.
2x + 9 > 14
2x > 5
x > `5/2`
Thus, for every x ∈ A, x must be greater than 2.5 Since all elements of A ({3, 5, 7, 9, 11, 12}) are greater than 2.5, we proceed to verify:
x ∈ A:
For x = 3, 2(3) + 9 = 15 > 14
For x = 5, 2(5) + 9 = 19 > 14
For x = 7, 2(7) + 9 = 23 > 14
For x = 9, 2(9) + 9 = 27 > 14
For x = 11, 2(11) + 9 = 31 > 14
For x = 12, 2(12) + 9 = 33 > 14
The inequality 2x + 9 > 14 holds for all x ∈ A.
The truth value of the statement is true.
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