Question

Prove the following : (∃x)(F(x)⋀(G(x)⋁(H(x)))→(∃x)(∃y)(F(x)⋀(G(y)⋁H(y)))

Prove the following : (∃x)(F(x)⋀(G(x)⋁(H(x)))→(∃x)(∃y)(F(x)⋀(G(y)⋁H(y)))

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Answer #2

INFORMAL PROOF.

(∃x)(F(x)⋀(G(x)⋁(H(x)))→(∃x)(∃y)(F(x)⋀(G(y)⋁H(y)))

A conditional statement P→Q is TRUE if whenever P is TRUE, Q is also TRUE.

Here, if for any x = xi U, the left side is TRUE, then F(xi) should be TRUE and any one of the G(xi) or H(xi) must also be true. From this we see that the Right Side expression will be true as for at least for y= x1, G(y) ⋁ H(y) will be TRUE.

So the given conditional expression will be TRUE in each case.

Hence, (∃x)(F(x)⋀(G(x)⋁(H(x)))→(∃x)(∃y)(F(x)⋀(G(y)⋁H(y))) will be TRUE for each x, and y.

answered by: anonymous
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