CSC 226 Chapter Notes - Chapter 1.3: Conditional Sentence, Truth Table, Contraposition
CSC226: Discrete Mathematics for Computer
Scientists
Zybooks 1.3: Conditional Statements
January 24, 2017
Conditional Propositions
• Conditional Operation: A special operation denoted with the symbol →.
• Conditional Proposition: A compound proposition that uses a conditional operation.
Written as p→q
o For example, if you wrote p→, the ou ead it out as if p, the
o It’s false if p is TRUE ad is FAL“E, otheise it’s tue.
o Hypothesis: The proposition to the LEFT of the → (p)
o Conclusion: The proposition to the RIGHT of the → (q)
o Truth table for the conditional operation
p
q
p→q
T
T
T
T
F
F
F
T
T
F
F
T
• The best way to think of conditional propositions is as a contract between the two
propositions.
• Use this analogy: If ou o Mr Ma’s la, the he ill pa ou.
o If you choose to mow the lawn and he pays you, the truth value of the
conditional proposition is TRUE
o If you choose to not mow the lawn, he can choose to pay you anyways or not
pay you- either way, the contract is not broken, so the truth value is still TRUE
o However, if you mow the lawn and he does NOT pay you, then the contract is
broken, rendering the conditional proposition with a truth value of FALSE.
• You can also use the previous analogy to describe ways of wording a conditional
operation
o If p, then q: If ou o M Ma’s la, the he ill pa ou
o If p, q: If ou o M Ma’s la, he ill pa ou
o q if p: Mr Mann will pay you if you mow his lawn
o p implies q: Moig M Ma’s la iplies that he ill pa ou
o p only if q: You ill o M Ma’s la ol if he pas ou
o p is sufficient for q: Moig M Ma’s la is suffiiet fo hi to pa ou
o q is necessary for p: M Ma’s paig ou is eessa fo ou to o his la.
Converse, Contrapositive, and Inverse
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Document Summary
Conditional propositions: conditional operation: a special operation denoted with the symbol , conditional proposition: a compound proposition that uses a conditional operation. Written as p q: for example, if you wrote p (cid:395), the(cid:374) (cid:455)ou (cid:396)ead it out as (cid:862)if p, the(cid:374) (cid:395)(cid:863) T: the best way to think of conditional propositions is as a contract between the two propositions, use this analogy: (cid:862)if (cid:455)ou (cid:373)o(cid:449) mr ma(cid:374)(cid:374)"s la(cid:449)(cid:374), the(cid:374) he (cid:449)ill pa(cid:455) (cid:455)ou. (cid:863) If you choose to mow the lawn and he pays you, the truth value of the conditional proposition is true. If p, then q: if (cid:455)ou (cid:373)o(cid:449) m(cid:396) ma(cid:374)(cid:374)"s la(cid:449)(cid:374), the(cid:374) he (cid:449)ill pa(cid:455) (cid:455)ou. Converse, contrapositive, and inverse: there are three conditional statements related to the compound proposition that are so (cid:272)o(cid:373)(cid:373)o(cid:374), the(cid:455) ha(cid:448)e thei(cid:396) o(cid:449)(cid:374) (cid:374)a(cid:373)es. We"ll des(cid:272)(cid:396)i(cid:271)e the(cid:373) usi(cid:374)g p q, with p: it is raining today, and q: the game is cancelled: converse: q p.
