Question

If A = {1, 2, 3, 4, 5} then which of the following is not true?

Options

• ∃ x ∈ A such that x + 3 = 8

• ∃ x ∈ A such that x + 2 < 9

• ∀ x ∈ A, x + 6 ≥ 9

• ∃ x ∈ A such that x + 6 < 10

Answer 1

∀ x ∈ A, x + 6 ≥ 9

Explanation:

Given set: A = {1, 2, 3, 4, 5}

1. ∃ x ∈ A such that x + 3 = 8

This means there exists at least one x in A for which:

x + 3 = 8

x = 8 − 3

x = 5

Since 5 ∈ A, this statement is true.

2. ∃ x ∈ A such that x + 2 < 9

This means there exists at least one x in A for which:

x + 2 < 9

Since for all values of x in A,

1 + 2 = 3,

2 + 2 = 4,

3 + 2 = 5,

4 + 2 = 6,

5 + 2 = 7

and all these values are less than 9, this statement is true.

3. ∀ x ∈ A, x + 6 ≥ 9

This means for all x in A,

x + 6 ≥ 9

Checking each value of x:

 1 + 6 = 7 (False, since 7 `≱ ` 9)

Since one counterexample (x = 1) disproves the statement, this statement is false.

4. ∃ x ∈ A such that x + 6 < 10

This means there exists at least one x in A for which:

x + 6 < 10

Checking each value of x:

1 + 6 = 7,

2 + 6 = 8,

3 + 6 = 9,

4 + 6 = 10,

5 + 6 = 11

Here, x = 1 satisfies 1 + 6 = 7 < 10

Since at least one x satisfies the condition, this statement is true.