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Consider a function f:R→Rf:R→Rf: R \to R, f(x)=2x+8f(x)=2x+8f(x) = 2x+8.

Determine the correct sequence to prove the following statement.

ff is surjective.

If a statement is not used in the proof, you have to choose "Not used".

Consider an arbitrary real number yy.

Assume f(x)=yf(x)=y.

2x+8=y2x+8=y

Consider two arbitrary real numbers aa and bb.

ff is injective

a=ba = b

2a+8=2b+82a+8=2b+8

Assume that f(a)=f(b)f(a)=f(b).

ff is surjective

x=y−82x = \frac{y-8}{2} exists

Consider a function f:R→Rf:R→Rf: R \to R, f(x)=2x+8f(x)=2x+8f(x) = 2x+8.

Determine the correct sequence to prove the following statement.

ff is injective.

If a statement is not used in the proof, you have to choose "Not used".

Consider an arbitrary real number yy.

Assume f(x)=yf(x)=y.

2x+8=y2x+8=y

Consider two arbitrary real numbers aa and bb.

ff is injective

a=ba = b

2a+8=2b+82a+8=2b+8

Assume that f(a)=f(b)f(a)=f(b).

ff is surjective

x=y−82x = \frac{y-8}{2} exists

Choose the main technique to prove the following statement:

3x+21=03x+21=03x+21=0 has an integer solution.

T6: Constructive proof to show ∃xP(x)≡T\exists x P(x) \equiv T

T7: Two-step proof to show ∃!xP(x)≡T\exists ! x P(x) \equiv T

T1: Direct proof to show (p→q)≡T(p \to q) \equiv T

T5: Proof by cases to show ((n⋁i=1pi)→q)≡T \displaystyle \left(\left(\bigvee_{i=1}^n p_i \right) \to q \right) \equiv T

Choose the main technique to prove the following statement:

It is false to claim that every integer is even.

T6: Constructive proof to show ∃xP(x)≡T∃xP(x)≡T\exists x P(x) \equiv T

T1: Direct proof to show (p→q)≡T(p→q)≡T(p \to q) \equiv T

T5: Proof by cases to show ((n⋁i=1pi)→q)≡T((n⋁i=1pi)→q)≡T \displaystyle \left(\left(\bigvee_{i=1}^n p_i \right) \to q \right) \equiv T

T7: Two-step proof to show ∃!xP(x)≡T∃!xP(x)≡T\exists ! x P(x) \equiv T