Constructing Proofs Using Symbolic Logic



Constructing Proofs Using Symbolic Logic

These proofs use only three reasons:

Given

Contrapositve of ______

Syllogism of _____ and _____

A true statement may be replaced by its contrapositive because they are equivalent as discussed on page 281 of the text. Example: If s → t then ~t → ~s where the symbol ~ means “not.”

The Law of Syllogism is given on page 95 and states that if p → q and q → r then p → r.

Example in flow proof form:

Given: p → q, ~r → ~q

Prove: p → r

Example in two-column form:

Statements Reasons

1. p → q 1. Given

2. ~r → ~q 2. Given

3. q → r 3. Contrapositive of 2

4. p → r 4. Syllogism of 1 and 3

Fill in the reasons in this flow proof.

Given: a → ~b, c → b, ~c → ~d

Prove: a → ~d

The same proof may be done correctly in more than one way. Fill in the reasons on the alternate proof for the same problem.

The major difference between the two proofs is the order in which the givens were written. Most of the time, a proof will be shorter if the “given” written first contains something with the hypothesis of what needs to be proven and the last written “given” contains something with the conclusion of what needs to be proven. In the problem above, the hypothesis of the “prove” is “a” so the given “a → ~b” is listed first. The conclusion of the “prove” is “~d” so the given

“~c → ~d” is written last.

On a separate sheet of paper, write a flow proof for each of the following.

1. Given: ~m → r, ~b → ~r

Prove: ~b → m

2. Given: p → ~r, ~q → ~t, ~t → r

Prove: p → q

3. Given: ~b → ~a, b → c, c → ~d

Prove: d → ~a

4. Given: ~s → r, s → ~t, w → t

Prove: w → r

5. Given: ~a → b, c → d, ~c → ~b

Prove: ~d → a

Additional practice:

1. Given: a → b, c → ~b

Prove: c → ~a

2. Given: a → ~b, ~a → c, d → ~c

Prove: b → ~d

3. Given: b → ~a, ~c → ~d, ~d → b

Prove: a → c

-----------------------

Given

Given

p → q

~r → ~q

Contrapositive

q → r

Syllogism

p → r

~c → ~d

c → b

a → ~b

d → c

d → b

b → ~a

d → ~a

a → ~d

a → ~d

a → ~c

~b → ~c

~c → ~d

c → b

a → ~b

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download