- सत्यम् / Satyam = Truth
(Difference between "True & Real" or "Reality vs. Truth" is another topic/question that might be suitable on english.SE or philosophy.SE)
- मिथ्या / Mithya = False / Unreal
According to Adi Shankara & Advaita Vedanta, world is unreal. The sanskrit word मिथ्या means unreal or false. It may also be used as unworthy.
ब्रह्म सत्यं जगन्मिथ्या :
Anything is called Sat (सत्) only if it exist (and remains constant) in all the three periods of time (past,current & future) i.e त्रिकालाबाध सत्य. But in case of world it isn't exist before origin & after dissolution, plus it doesn't remain constant in three periods of time. This way it (world) is called मिथ्या=false (असत्). But Brahman(ब्रह्म) is सत् as I've discussed here. सत्यं ज्ञानमनन्तं ब्रह्म (Taittariya Upanishad 2-1).
In the word "जगत् मिथ्या", जगत् is also taken as माया (illusion) which can be "सत्" for who realizing it because of ignorance. Who tries to understand the maya, can't call it सत् as discussed in above para and neither call it असत् as realizing it and hence call it अनिर्वचनीय (Inexpressible). But For who knows the ultimate truth, there is no such thing and it becomes मिथ्या=unworthy (insignificant).
Now, going to the philosophy of Advaita:
Gaudapada (Gaudapada was the grand guru of Adi Shankaracharya) karika on Mandukya Upanishad (also known as Māndūkya-kārikā) can be considered as the first/oldest source for Advaita. In chapter 2 (वैतथ्य प्रकरण), मिथ्यात्व (unreality/illusion) of the world is demonstrated by logical arguments.
You find many-times the Sanskrit word "वैतथ्य" (falseness) there but as this question deals with the word "मिथ्या", let me quote one verse:
सप्रयोजनता तेषां स्वप्ने विप्रतिपद्यते ।
तस्मादाद्यन्तवत्त्वेन मिथ्यैव खलु ते स्मृताः ॥
Above verse is found two-times (2-7 and 4-32) which concludes both state i.e जाग्रत (Waking) and स्वप्न (dreaming) are unreal and illusory. The complementaries of Adi Shankaracharya also say this.
Another interesting and clear verse is:
धर्मा य इति जायन्ते जायन्ते ते न तत्त्वतः ।
जन्म मायोपमं तेषां सा च माया न विद्यते ॥
on which Adi Shankara's commentaries states:
If all existing is due to Maya (माया), then isn't maya is real/actual? No; सा च माया न विद्यते Maya is not exist. It is only the name of (non-exist) unreal thing.
So, The conclusion is: According to Advaita, world is not exist, it is unreal. (I recommend reading the Mandukya Upanishad with Gaudpad Karika for clear understanding of this).
Edit : Including verse 231-238 from Vivekachudamani (English translation from here:
ब्रह्मैवेदं विश्वमित्येव वाणी श्रौती ब्रूतेऽथर्वनिष्ठा वरिष्ठा ।
तस्मादेतद्ब्रह्ममात्रं हि विश्वं नाधिष्ठानाद्भिन्नताऽऽरोपितस्य ॥
This universe is verily Brahman – such is the august pronouncement of the Atharva Veda. Therefore this universe is nothing but Brahman, for that which is superimposed (on something) has no separate existence from its substratum.
सत्यं यदि स्याज्जगदेतदात्मनोऽ न तत्त्वहानिर्निगमाप्रमाणता ।
असत्यवादित्वमपीशितुः स्या- न्नैतत्त्रयं साधु हितं महात्मनाम् ॥
If the universe, as it is, be real, there would be no cessation of the dualistic element, the scriptures would be falsified, and the Lord Himself would be guilty of an untruth. None of these three is considered either desirable or wholesome by the noble-minded.
ईश्वरो वस्तुतत्त्वज्ञो न चाहं तेष्ववस्थितः । न च मत्स्थानि भूतानीत्येवमेव व्यचीक्लृपत् ॥
The Lord, who knows the secret of all things has supported this view in the words: "But I am not in them" … "nor are the beings in Me".
यदि सत्यं भवेद्विश्वं सुषुप्तामुपलभ्यताम् । यन्नोपलभ्यते किञ्चिदतोऽसत्स्वप्नवन्मृषा ॥
If the universe be true, let it then be perceived in the state of deep sleep also. As it is not at all perceived, it must be unreal and false, like dreams.
अतः पृथङ्नास्ति जगत्परात्मनः पृथक्प्रतीतिस्तु मृषा गुणादिवत् ।
आरोपितस्यास्ति किमर्थवत्ताऽ- धिष्ठानमाभाति तथा भ्रमेण ॥ २३५॥
Therefore the universe does not exist apart from the Supreme Self; and the perception of its separateness is false like the qualities (of blueness etc., in the sky). Has a superimposed attribute any meaning apart from its substratum ? It is the substratum which appears like that through delusion.
भ्रान्तस्य यद्यद्भ्रमतः प्रतीतं ब्रह्मैव तत्तद्रजतं हि शुक्तिः ।
इदन्तया ब्रह्म सदैव रूप्यते त्वारोपितं ब्रह्मणि नाममात्रम् ॥
Whatever a deluded man perceives through mistake, is Brahman and Brahman alone: The silver is nothing but the mother-of-pearl. It is Brahman which is always considered as this universe, whereas that which is superimposed on the Brahman, viz. the universe, is merely a name.
अतः परं ब्रह्म सदद्वितीयं विशुद्धविज्ञानघनं निरञ्जनम् ।
प्राशान्तमाद्यन्तविहीनमक्रियं निरन्तरानन्दरसस्वरूपम् ॥
निरस्तमायाकृतसर्वभेदं नित्यं सुखं निष्कलमप्रमेयम् ।
अरूपमव्यक्तमनाख्यमव्ययं ज्योतिः स्वयं किञ्चिदिदं चकास्ति ॥
Hence whatever is manifested, viz. this universe, is the Supreme Brahman Itself, the Real, the One without a second, pure, the Essence of Knowledge, taintless, serene, devoid of beginning and end, beyond activity, the Essence of Bliss Absolute – transcending all the diversities created by Maya or Nescience, eternal, ever beyond the reach of pain, indivisible, immeasurable, formless, undifferentiated, nameless, immutable, self-luminous.
Which clearly demonstrates that world/universe is unreal!
Note: Without doing more arguments we should focus on attaining the ultimate truth i.e Aatman (आत्मा) /Brahman (ब्रह्म). [Mundaka Upanishad 1-3 : knowing what all this things can be known], [Brihadaranyak 2-8-5]